The Concept of Midpoints and Section Points

Oct 13, 2025 by pinkandtrash.com 49 views

Understanding the Problem: A Geometric Approach to Finding Point Coordinates

In geometry, the concept of a point being a certain fraction of the way from one point to another is a fundamental idea. Given two points, A and B, and a point P that is 1/3 of the way from A to B, we are tasked with finding the coordinates of point P. In this article, we will delve into the world of coordinate geometry and explore the steps necessary to determine the values of p and q.

The Concept of Midpoints and Section Points

To begin, let's consider the concept of midpoints and section points. A midpoint is a point that divides a line segment into two equal parts. On the other hand, a section point is a point that divides a line segment into two parts, where the ratio of the lengths of the two parts is a given fraction. In this case, we are dealing with a section point that divides the line segment AB into three equal parts.

The Formula for a Section Point

The formula for a section point that divides a line segment into two parts, where the ratio of the lengths of the two parts is a given fraction, is as follows:

P = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)

where (x1, y1) and (x2, y2) are the coordinates of the two points, and m and n are the ratios of the lengths of the two parts.

Applying the Formula to the Given Problem

Now, let's apply the formula to the given problem. We are given that point P is 1/3 of the way from A to B. Using the formula, we can write:

P = \left(\frac{1/3x_2 + 2/3x_1}{1/3 + 2/3}, \frac{1/3y_2 + 2/3y_1}{1/3 + 2/3}\right)

Simplifying the expression, we get:

P = \left(\frac{x_2 + 2x_1}{3}, \frac{y_2 + 2y_1}{3}\right)

Finding the Values of p and q

Now that we have the coordinates of point P, we can find the values of p and q. Since point P is 1/3 of the way from A to B, we can write:

p = \frac{x_2 + 2x_1}{3}

q = \frac{y_2 + 2y_1}{3}

In conclusion, we have successfully found the values of p and q using the formula for a section point. By applying the formula to the given problem, we were able to determine the coordinates of point P and find the values of p and q. This problem demonstrates the importance of understanding the concept of midpoints and section points in geometry, and how they can be used to solve problems involving coordinate geometry.

Suppose we have two points A(1, 2) and B(4, 6), and we want to find the coordinates of point P that is 1/3 of the way from A to B. Using the formula, we can find the coordinates of point P as follows:

P = \left(\frac{1/3(4) + 2/3(1)}{1/3 + 2/3}, \frac{1/3(6) + 2/3(2)}{1/3 + 2/3}\right)

Simplifying the expression, we get:

P = \left(\frac{4 + 2}{3}, \frac{6 + 4}{3}\right)

P = \left(\frac{6}{3}, \frac{10}{3}\right)

P = (2, \frac{10}{3})

Therefore, the coordinates of point P are (2, 10/3).

The final answer is:

p = \frac{x_2 + 2x_1}{3}

q = \frac{y_2 + 2y_1}{3}$$<br/> **Q&A: Understanding the Problem of Finding Point Coordinates**

In our previous article, we explored the concept of midpoints and section points in geometry, and how they can be used to solve problems involving coordinate geometry. We also derived the formula for a section point that divides a line segment into two parts, where the ratio of the lengths of the two parts is a given fraction. In this article, we will answer some frequently asked questions related to the problem of finding point coordinates.

Q: What is the formula for a section point?

A: The formula for a section point that divides a line segment into two parts, where the ratio of the lengths of the two parts is a given fraction, is as follows:

P = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right) </span></p> <p>where (x1, y1) and (x2, y2) are the coordinates of the two points, and m and n are the ratios of the lengths of the two parts.</p> <h2><strong>Q: How do I apply the formula to find the coordinates of a point that is a certain fraction of the way from two given points?</strong></h2> <p>A: To apply the formula, you need to substitute the values of the two points and the fraction into the formula. For example, if you want to find the coordinates of a point that is 1/3 of the way from A(1, 2) to B(4, 6), you would substitute the values as follows:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo>+</mo><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo>+</mo><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">P = \left(\frac{1/3(4) + 2/3(1)}{1/3 + 2/3}, \frac{1/3(6) + 2/3(2)}{1/3 + 2/3}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1/3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2/3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1/3</span><span class="mopen">(</span><span class="mord">4</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2/3</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1/3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2/3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1/3</span><span class="mopen">(</span><span class="mord">6</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2/3</span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <p>Simplifying the expression, you get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mrow><mn>4</mn><mo>+</mo><mn>2</mn></mrow><mn>3</mn></mfrac><mo separator="true">,</mo><mfrac><mrow><mn>6</mn><mo>+</mo><mn>4</mn></mrow><mn>3</mn></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">P = \left(\frac{4 + 2}{3}, \frac{6 + 4}{3}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mn>6</mn><mn>3</mn></mfrac><mo separator="true">,</mo><mfrac><mn>10</mn><mn>3</mn></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">P = \left(\frac{6}{3}, \frac{10}{3}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">10</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mfrac><mn>10</mn><mn>3</mn></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P = (2, \frac{10}{3}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">10</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></span></p> <h2><strong>Q: What is the significance of the ratio of the lengths of the two parts in the formula?</strong></h2> <p>A: The ratio of the lengths of the two parts in the formula is used to determine the coordinates of the point that divides the line segment into two parts. The ratio is used to calculate the x and y coordinates of the point, which are then used to determine the coordinates of the point.</p> <h2><strong>Q: Can I use the formula to find the coordinates of a point that is a negative fraction of the way from two given points?</strong></h2> <p>A: Yes, you can use the formula to find the coordinates of a point that is a negative fraction of the way from two given points. To do this, you need to substitute the negative fraction into the formula. For example, if you want to find the coordinates of a point that is -1/3 of the way from A(1, 2) to B(4, 6), you would substitute the values as follows:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mrow><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo>+</mo><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo>+</mo><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">P = \left(\frac{-1/3(4) + 2/3(1)}{-1/3 + 2/3}, \frac{-1/3(6) + 2/3(2)}{-1/3 + 2/3}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1/3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2/3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1/3</span><span class="mopen">(</span><span class="mord">4</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2/3</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1/3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2/3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1/3</span><span class="mopen">(</span><span class="mord">6</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2/3</span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <p>Simplifying the expression, you get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mrow><mo>−</mo><mn>4</mn><mo>+</mo><mn>2</mn></mrow><mrow><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo>+</mo><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mo>−</mo><mn>6</mn><mo>+</mo><mn>4</mn></mrow><mrow><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo>+</mo><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">P = \left(\frac{-4 + 2}{-1/3 + 2/3}, \frac{-6 + 4}{-1/3 + 2/3}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1/3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2/3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1/3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2/3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mrow><mo>−</mo><mn>2</mn></mrow><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mo>−</mo><mn>2</mn></mrow><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">P = \left(\frac{-2}{1/3}, \frac{-2}{1/3}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1/3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1/3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>6</mn><mo separator="true">,</mo><mo>−</mo><mn>6</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P = (-6, -6) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">6</span><span class="mclose">)</span></span></span></span></span></p> <h2><strong>Q: Can I use the formula to find the coordinates of a point that is a fraction of the way from two given points, where the fraction is a decimal?</strong></h2> <p>A: Yes, you can use the formula to find the coordinates of a point that is a fraction of the way from two given points, where the fraction is a decimal. To do this, you need to substitute the decimal fraction into the formula. For example, if you want to find the coordinates of a point that is 0.5 of the way from A(1, 2) to B(4, 6), you would substitute the values as follows:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mrow><mn>0.5</mn><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>+</mo><mn>0.5</mn><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>0.5</mn><mo>+</mo><mn>0.5</mn></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mn>0.5</mn><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo><mo>+</mo><mn>0.5</mn><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mn>0.5</mn><mo>+</mo><mn>0.5</mn></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">P = \left(\frac{0.5(4) + 0.5(1)}{0.5 + 0.5}, \frac{0.5(6) + 0.5(2)}{0.5 + 0.5}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">0.5</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.5</span><span class="mopen">(</span><span class="mord">4</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">0.5</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">0.5</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.5</span><span class="mopen">(</span><span class="mord">6</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">0.5</span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <p>Simplifying the expression, you get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mrow><mn>2</mn><mo>+</mo><mn>0.5</mn></mrow><mn>1</mn></mfrac><mo separator="true">,</mo><mfrac><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow><mn>1</mn></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">P = \left(\frac{2 + 0.5}{1}, \frac{3 + 1}{1}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">0.5</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mn>2.5</mn><mn>1</mn></mfrac><mo separator="true">,</mo><mfrac><mn>4</mn><mn>1</mn></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">P = \left(\frac{2.5}{1}, \frac{4}{1}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2.5</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2.5</mn><mo separator="true">,</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P = (2.5, 4) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2.5</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span><span class="mclose">)</span></span></span></span></span></p> <p>In conclusion, we have answered some frequently asked questions related to the problem of finding point coordinates. We have discussed the formula for a section point, how to apply the formula to find the coordinates of a point that is a certain fraction of the way from two given points, and how to use the formula to find the coordinates of a point that is a negative fraction or a decimal fraction of the way from two given points. We hope that this article has been helpful in understanding the concept of midpoints and section points in geometry.</p>