What is the length of a leg of an isosceles right triangle whose hypotenuse measures 6 inches?
Let [tex]$c$[/tex] represent the value of the hypotenuse. If the hypotenuse is [tex]$c=a \sqrt{2}$[/tex], then [tex]$6=a \sqrt{2}$[/tex]. In inches, what is the value of [tex]$a$[/tex]?
A. 3
B. [tex]$3 \sqrt{2}$[/tex]
C. 6
D. [tex]$6 \sqrt{2}$[/tex]
Answer (1)
Given an isosceles right triangle with hypotenuse c = 6 inches.
Use the relationship c = a 2 , where a is the length of a leg.
Solve for a by dividing both sides by 2 : a = 2 6 .
Rationalize the denominator: a = 2 6 2 = 3 2 .
The length of a leg is 3 2 .
Explanation
Problem Analysis and Setup We are given an isosceles right triangle with a hypotenuse of 6 inches. We need to find the length of one of its legs. Let's denote the length of a leg as a . In an isosceles right triangle, the relationship between the leg length a and the hypotenuse c is given by the Pythagorean theorem: a 2 + a 2 = c 2 , which simplifies to 2 a 2 = c 2 . Alternatively, we can express the hypotenuse as c = a \[ 0.5 e x ] s q r t 2 .
Substitute the given value We are given that the hypotenuse c = 6 inches. We can use the formula c = a \[ 0.5 e x ] s q r t 2 to solve for a . Substituting the value of c , we get:
6 = a \[ 0.5 e x ] s q r t 2
Isolate a To find a , we need to isolate it by dividing both sides of the equation by 2 :
a = 2 6
Rationalize the denominator To rationalize the denominator, we multiply both the numerator and the denominator by 2 :
a = 2 6 × 2 2 = 2 6 2
Simplify the expression Now, we simplify the fraction:
a = 2 6 2 = 3 2
So, the length of a leg of the isosceles right triangle is 3 2 inches.
Final Answer Therefore, the length of a leg of the isosceles right triangle is 3 2 inches.
Examples
Isosceles right triangles are commonly found in construction and design. For example, when building a ramp that needs to have a 45-degree angle, the sides forming the right angle are equal in length. If you know the length of the ramp (the hypotenuse), you can calculate the length of the sides using the formula we just derived. This ensures the ramp has the correct angle and dimensions, which is crucial for safety and functionality.
