Applying the Properties to the Problem

Oct 13, 2025 by pinkandtrash.com 74 views

Solving a Classic Geometry Problem: Finding the Length of the Hypotenuse in a 45°-45°-90° Triangle

In geometry, a 45°-45°-90° triangle is a special right-angled triangle with two equal acute angles, each measuring 45°. This type of triangle has several unique properties that make it an essential concept in mathematics. In this article, we will explore the properties of a 45°-45°-90° triangle and use it to solve a classic problem: finding the length of the hypotenuse when each leg measures 12 cm.

Understanding the Properties of a 45°-45°-90° Triangle

A 45°-45°-90° triangle has several key properties that make it easy to work with:

Two equal acute angles: Both acute angles in a 45°-45°-90° triangle measure 45°.
One right angle: The third angle in a 45°-45°-90° triangle is a right angle, measuring 90°.
Two equal legs: The two legs of a 45°-45°-90° triangle are equal in length.
Hypotenuse is √2 times the length of a leg: The length of the hypotenuse in a 45°-45°-90° triangle is √2 times the length of a leg.

Applying the Properties to the Problem

Now that we understand the properties of a 45°-45°-90° triangle, let's apply them to the problem at hand. We are given that each leg of the triangle measures 12 cm, and we need to find the length of the hypotenuse.

Using the property that the hypotenuse is √2 times the length of a leg, we can calculate the length of the hypotenuse as follows:

Hypotenuse = √2 × Leg

Hypotenuse = √2 × 12 cm

Hypotenuse = 12√2 cm

Therefore, the length of the hypotenuse in a 45°-45°-90° triangle with each leg measuring 12 cm is 12√2 cm.

In this article, we explored the properties of a 45°-45°-90° triangle and used them to solve a classic problem: finding the length of the hypotenuse when each leg measures 12 cm. We learned that the hypotenuse is √2 times the length of a leg, and we applied this property to calculate the length of the hypotenuse. The result is 12√2 cm, which is the correct answer to the problem.

What is a 45°-45°-90° triangle? A 45°-45°-90° triangle is a special right-angled triangle with two equal acute angles, each measuring 45°.
What are the properties of a 45°-45°-90° triangle? The properties of a 45°-45°-90° triangle include two equal acute angles, one right angle, two equal legs, and a hypotenuse that is √2 times the length of a leg.
How do I find the length of the hypotenuse in a 45°-45°-90° triangle? To find the length of the hypotenuse in a 45°-45°-90° triangle, you can use the property that the hypotenuse is √2 times the length of a leg.

Geometry: Geometry is the branch of mathematics that deals with the study of shapes, sizes, and positions of objects.
Trigonometry: Trigonometry is the branch of mathematics that deals with the study of triangles, particularly right-angled triangles.
Right-angled triangles: Right-angled triangles are triangles with one right angle, measuring 90°.

Hypotenuse: The hypotenuse is the longest side of a right-angled triangle, opposite the right angle.
Leg: A leg is one of the two sides of a right-angled triangle that meet at a right angle.
Right angle: A right angle is an angle that measures 90°.
√2: √2 is the square root of 2, approximately equal to 1.414.
45°-45°-90° Triangle Q&A: Frequently Asked Questions and Answers ====================================================================

In our previous article, we explored the properties of a 45°-45°-90° triangle and used them to solve a classic problem: finding the length of the hypotenuse when each leg measures 12 cm. In this article, we will answer some of the most frequently asked questions about 45°-45°-90° triangles.

Q: What is a 45°-45°-90° triangle?

A: A 45°-45°-90° triangle is a special right-angled triangle with two equal acute angles, each measuring 45°.

Q: What are the properties of a 45°-45°-90° triangle?

A: The properties of a 45°-45°-90° triangle include two equal acute angles, one right angle, two equal legs, and a hypotenuse that is √2 times the length of a leg.

Q: How do I find the length of the hypotenuse in a 45°-45°-90° triangle?

A: To find the length of the hypotenuse in a 45°-45°-90° triangle, you can use the property that the hypotenuse is √2 times the length of a leg.

Q: What is the relationship between the legs and the hypotenuse in a 45°-45°-90° triangle?

A: In a 45°-45°-90° triangle, the legs are equal in length, and the hypotenuse is √2 times the length of a leg.

Q: Can I use the Pythagorean theorem to find the length of the hypotenuse in a 45°-45°-90° triangle?

A: Yes, you can use the Pythagorean theorem to find the length of the hypotenuse in a 45°-45°-90° triangle. However, it is often easier to use the property that the hypotenuse is √2 times the length of a leg.

Q: What are some real-world applications of 45°-45°-90° triangles?

A: 45°-45°-90° triangles have many real-world applications, including:

Construction: 45°-45°-90° triangles are used in construction to calculate the length of roof rafters and other diagonal members.
Engineering: 45°-45°-90° triangles are used in engineering to calculate the stress and strain on materials.
Design: 45°-45°-90° triangles are used in design to create symmetrical and balanced compositions.

Q: Can I use a 45°-45°-90° triangle to solve other types of problems?

A: Yes, you can use a 45°-45°-90° triangle to solve other types of problems, including:

Right triangle problems: You can use a 45°-45°-90° triangle to solve right triangle problems, such as finding the length of the hypotenuse or the length of a leg.
Oblique triangle problems: You can use a 45°-45°-90° triangle to solve oblique triangle problems, such as finding the length of a side or the measure of an angle.

In this article, we answered some of the most frequently asked questions about 45°-45°-90° triangles. We hope that this article has been helpful in clarifying the properties and applications of 45°-45°-90° triangles.

What is a 45°-45°-90° triangle? A 45°-45°-90° triangle is a special right-angled triangle with two equal acute angles, each measuring 45°.
What are the properties of a 45°-45°-90° triangle? The properties of a 45°-45°-90° triangle include two equal acute angles, one right angle, two equal legs, and a hypotenuse that is √2 times the length of a leg.
How do I find the length of the hypotenuse in a 45°-45°-90° triangle? To find the length of the hypotenuse in a 45°-45°-90° triangle, you can use the property that the hypotenuse is √2 times the length of a leg.

Geometry: Geometry is the branch of mathematics that deals with the study of shapes, sizes, and positions of objects.
Trigonometry: Trigonometry is the branch of mathematics that deals with the study of triangles, particularly right-angled triangles.
Right-angled triangles: Right-angled triangles are triangles with one right angle, measuring 90°.

Hypotenuse: The hypotenuse is the longest side of a right-angled triangle, opposite the right angle.
Leg: A leg is one of the two sides of a right-angled triangle that meet at a right angle.
Right angle: A right angle is an angle that measures 90°.
√2: √2 is the square root of 2, approximately equal to 1.414.