If the altitude of an isosceles right triangle has a length of x units, what is the length of one leg of the triangle in terms of [tex]$x$[/tex]?
A. [tex]$x$[/tex] units
B. [tex]$x \sqrt{2}$[/tex] units
C. [tex]$x \sqrt{3}$[/tex] units
D. [tex]$2x$[/tex] units
Answer (1)
Define the length of the leg of the isosceles right triangle as a .
Express the area of the triangle in two ways: using the legs and using the hypotenuse and altitude.
Set the two area expressions equal to each other and solve for a in terms of x .
The length of one leg is x 2 .
Explanation
Problem Analysis Let's analyze the problem. We have an isosceles right triangle, which means it has two equal sides and a right angle. The altitude to the hypotenuse has length x . We want to find the length of one of the legs in terms of x .
Calculations Let a be the length of each leg of the isosceles right triangle. The hypotenuse has length a 2 by the Pythagorean theorem. The area of the triangle is 2 1 a 2 . We can also express the area as 2 1 × hypotenuse × altitude to hypotenuse = 2 1 ( a 2 ) x . Setting these equal gives 2 1 a 2 = 2 1 a x 2 a 2 = a x 2 Since a = 0 , we can divide by a to get a = x 2
Final Answer Therefore, the length of one leg of the isosceles right triangle is x 2 units.
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 length of the legs are equal. If you know the altitude to the hypotenuse, you can easily calculate the length of the legs using the formula we derived. This helps in ensuring the ramp has the correct dimensions and angle.
