Question

An isosceles right triangle and a square have the same area. If the length of the hypotenuse of the right triangle is 6 √ 2, what is the area of the square?

f) 12
g) 12+6 √ 2
h) 18
j) 24+6 √ 2
k) 36

Answer 1

An isosceles right triangle is a 45°-45°-90° triangle. 45°-45°-90° triangles' hypotenuses are always equal to their leg length times the square root of two.

For 45°-45°-90° triangles:

Leg = x

Hypotenuse = x√(2)

The length of the hypotenuse is given (6√(2)), so we can easily find the length of its legs. If we divide the hypotenuse by √(2), we are left with 6. This is the leg length.

To find the area of the triangle (and thus the area of the square), use the formula for area of a triangle, A = bh/2. Our b, base, is 6, and our h, height, is also 6. Plug these values in accordingly and solve.

A = 6 * 6/2

A = 36/2

A = 18

The area of the triangle (and that of the square) is 18 units squared.

Answer:

The area of the square is 18 units².

h) 18

Answer 2

Isosceles right triangles are a special right triangle because it follows the 45-45-90 rule. If one leg was x, then the other leg was also x and the hypotenuse was x√2.

Using that information, we can confirm that the legs of the triangle are 2 since 6√2 divided by √2 is 6.

Now that we have the legs of the triangle, we can use the formula for the area of the triangle to solve: (remember that area = (bh)/2)

`A=(6*6)/(2)`

Firstly, multiply the numerator:
`A=(36)/(2)`

Next, divide, and your answer will be A = 18. Since the square and the right triangle have the same area, your answer is 18, or h.