Key Concepts
- Find hypotenuse length in a 45°-45°-90° triangle
Special Right Triangles
There are two special right triangles with angles measures as 45°, 45°, 90° degrees and 30°, 60°, 90° degrees. The sides of these triangles are in particular ratios and are known as Pythagorean triplets.
45°-45°-90° Triangle
In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is √2
times the length of a leg.
![45°-45°-90° Triangle](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/1-9.png)
![45°-45°-90° Triangle](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/2-8.png)
Find hypotenuse length in a 45°-45°-90° triangle
Hypotenuse
A hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle.
Example 1:
Find the length of the hypotenuse.
![Example 1:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2646.png)
Solution:
By the Triangle Sum Theorem, the measure of the third angle must be 45°.
Then the triangle is a 45°-45°-90° triangle, so by theorem, the hypotenuse is
√2 times as long as each leg.
Hypotenuse = leg ×√2 45°-45°-90° Triangle Theorem
= 7√2 (Substitute)
Hypotenuse = leg√𝟐
Example 2:
Find the length of the hypotenuse.
![Example 2:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2644.png)
Solution:
By the 45° -45°-90° triangle theorem, the length of the hypotenuse is the length of a leg times √2
hypotenuse = leg × √2
= 3 × √2
The length of the Hypotenuse is 3√2
Example 3:
Find leg lengths in a 45°-45°-90° triangle.
![Example 3:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2646.png)
Solution:
By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a 45°-45°-90° triangle.
hypotenuse = leg × √2 ( 45°-45°-90° Triangle Theorem )
2√2 = x √2 (Substitute)
2√2/√2 = x√2/√2
2 = x (Simplify)
x = 2
Example 4:
Find the length of a leg of 45° -45°- 90° triangle with a hypotenuse of length 10.
![Example 4:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2650.png)
Solution:
hypotenuse = leg ×√2
10 = √x ×2
x = 10/√2
x = 10/√2 * √2/√2
x = 10*√2/2
x = 5 √2
Exercise
- Identify the legs and hypotenuse of the triangle.
![Identify the legs and hypotenuse of the triangle.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2652.png)
- 45°-45°-90° TRIANGLES: Find the value of x. Write your answer in the simplest radical form.
![45°-45°-90° TRIANGLES: Find the value of x. Write your answer in the simplest radical form.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2653.png)
- The square tile shown has painted corners in the shape of congruent 45°-45°-90° triangles. What is the value of x? What is the side length of the tile?
![The square tile shown has painted corners in the shape of congruent 45°-45°-90° triangles. What is the value of x? What is the side length of the tile?](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2655.png)
- Copy and complete the table.
![Copy and complete the table.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2654.png)
![Copy and complete the table.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2656.png)
- Is it possible to build a triangle using the given side lengths?
4, 4, and 7
- Find the values of the variable(s). Write your answer(s) in the simplest radical form.
![Find the values of the variable(s). Write your answer(s) in the simplest radical form.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2657.png)
- Find the length of a leg of 45° -45°- 90° triangle with a hypotenuse of length 6.
![Find the length of a leg of 45° -45°- 90° triangle with a hypotenuse of length 6.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2658.png)
- Find the length of the hypotenuse in the 45° – 45° – 90° triangle. Write your answer in radical form.
![Find the length of the hypotenuse in the 45° – 45° – 90° triangle. Write your answer in radical form.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2659.png)
- Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22.
![Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2660.png)
- Find the length of the hypotenuse.
![Find the length of the hypotenuse.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2661.png)
Concept Map
![Concept Map](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2651.png)
What have we learned
- Understand special right triangles.
- Understand 45°- 45° -90° Triangle theorem
- Understand how to find the length of the hypotenuse.
- Understand how to find the lengths of the legs in the triangle.
Related topics
Addition and Multiplication Using Counters & Bar-Diagrams
Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]
Read More >>Dilation: Definitions, Characteristics, and Similarities
Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]
Read More >>How to Write and Interpret Numerical Expressions?
Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […]
Read More >>System of Linear Inequalities and Equations
Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]
Read More >>Other topics
![demo](https://a.storyblok.com/f/128066/544x376/641160f2e6/learn_toast.webp)
Comments: