What is the area of ABCDE?

[AAPL]

What is the area of ABCDE?

$$A.108$$
$$B.126$$
$$C.\ 144-6\sqrt{3}$$
$$D.\ 144-18\sqrt{3}$$
$$E.\ 144-36\sqrt{3}$$

The OA is E.

Experts, can I say that ADE is an equilateral triangle with side 12? Then I can get the area of this equilateral triangle and finally the area of ABCDE will be,
$$A_{ABCDE}=A_{ABCD}-A_{ADE}$$
I appreciate if any expert explain this PS question for me. Thank you so much.

[GMATGuruNY]

What is the area of ABCDE?

$$A.108$$
$$B.126$$
$$C.\ 144-6\sqrt{3}$$
$$D.\ 144-18\sqrt{3}$$
$$E.\ 144-36\sqrt{3}$$

The OA is E.

Experts, can I say that ADE is an equilateral triangle with side 12? Then I can get the area of this equilateral triangle and finally the area of ABCDE will be,
$$A_{ABCDE}=A_{ABCD}-A_{ADE}$$
I appreciate if any expert explain this PS question for me. Thank you so much.

Your approach is perfect.
Area of an equilateral triangle with side s = (s²/4)√3.

In the figure above:
Area of square ABCD = s² = 12² = 144.
Area of equilateral triangle ADE = (s²/4)√3 = (12²/4)√3 = 36√3.
Shaded region ABCDE = (square ABCD) - (triangle ADE) = 144 - 36√3.

The correct answer is E.

Alternate approach:
√3 ≈ 1.7.
In the figure above, triangle ADE constitutes little less than half of square ABCD, implying that shaded region ABCDE constitutes a little MORE than half of square ABCD.
Thus, the area of ABCDE must be a little more than 72.
Of the five answer choices, only E is viable:
144 - 36√3 ≈ 144 - (36 * 1.7) ≈ 144 - 60 = 84.

To understand why triangle ADE must constitute less than half of square ABCD, check my posts here:
https://www.beatthegmat.com/area-of-pqr-t115199.html

[Brent@GMATPrepNow]

What is the area of ABCDE?

$$A.108$$
$$B.126$$
$$C.\ 144-6\sqrt{3}$$
$$D.\ 144-18\sqrt{3}$$
$$E.\ 144-36\sqrt{3}$$

[spoiler=]

area.PNG

[/spoiler][/quote]

Let's add a line from A to D:


When we do this, we see that ABCD is a square, and ADE is an equilateral triangle.

We can also write: area of ABCDE = (area of square ABCD) - (area of equilateral triangle ADE)

Useful formula: Area of equilateral triangle = (√3)(side²)/4

We get: ABCDE = (12²) - ((√3)(12²)/4)

Simplify: ABCDE = 144 - 36√3

Answer: E

Cheers,
Brent