## In the figure above, if triangles $$ABC, ACD,$$ and $$ADE$$ are isosceles right triangles and the area of

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### In the figure above, if triangles $$ABC, ACD,$$ and $$ADE$$ are isosceles right triangles and the area of

by Vincen » Sat Dec 04, 2021 7:34 am

00:00

A

B

C

D

E

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In the figure above, if triangles $$ABC, ACD,$$ and $$ADE$$ are isosceles right triangles and the area of $$\triangle ABC$$ is $$6,$$ then the area of $$\triangle ADE$$ is

A. $$18$$
B. $$24$$
C. $$36$$
D. $$12\sqrt2$$
E. $$24\sqrt2$$

Source: Official Guide

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### Re: In the figure above, if triangles $$ABC, ACD,$$ and $$ADE$$ are isosceles right triangles and the area of

by [email protected] » Sun Dec 05, 2021 6:26 am
Vincen wrote:
Sat Dec 04, 2021 7:34 am
2019-04-26_1753.png

In the figure above, if triangles $$ABC, ACD,$$ and $$ADE$$ are isosceles right triangles and the area of $$\triangle ABC$$ is $$6,$$ then the area of $$\triangle ADE$$ is

A. $$18$$
B. $$24$$
C. $$36$$
D. $$12\sqrt2$$
E. $$24\sqrt2$$

Source: Official Guide
In general, isosceles right triangles have the following properties.

Notice that the hypotenuse = (√2)(length of one leg)

So, let's label the sides of the BLUE triangle as follows:

This means each leg of the RED triangle has length (√2)(x)

The hypotenuse of an isosceles right triangle = (√2)(length of one leg) . . .

. . . so the length of the hypotenuse = (√2)(√2)(x) = 2x

This means each leg of the GREEN triangle has length 2x

What is the area of ΔADE?

Area of triangle = (base)(height)/2
So, area of ΔADE = (2x)(2x)/2 = 2x²

So, what is the value of 2x²?

GIVEN: the area of ΔABC is 6
ΔABC is the BLUE triangle we started with.
We can write: (x)(x)/2 = 6
Simplify: x²/2 = 6, which means x² = 12

This means the area of ΔADE = 2x² = 2(12) = 24