In the right triangle \(ABC\) above, the length of \(AB\) is

[Vincen]

In the right triangle \(ABC\) above, the length of \(AB\) is \(53\sqrt{53}\). What is the area of triangle \(ABC\)?

(1) \(DBC\) is an equilateral triangle.
(2) \(ABD\) is an isosceles triangle.

[spoiler]OA=A[/spoiler]

Source: GMAT Club Tests

[Jay@ManhattanReview]

In the right triangle \(ABC\) above, the length of \(AB\) is \(53\sqrt{53}\). What is the area of triangle \(ABC\)?

(1) \(DBC\) is an equilateral triangle.
(2) \(ABD\) is an isosceles triangle.

[spoiler]OA=A[/spoiler]

Source: GMAT Club Tests

To know the area of ∆ABC, we must know the values of AB and BC. We are given that AB = 53√53. Thus, we have to get the value of BC.

Let's take each statement one by one.

(1) \(DBC\) is an equilateral triangle.

Since ∆DBC is an equilateral triangle, we have BC = CD = BD = say x. Also /_C = /_ABD = 90º - 60º = 30º, thus, /_A = 180 - 90 - 60 = 30º.

In ∆ABD, we have /_ABD = /_A = 30º; thus, BD = AD = x

=> AC = x + x = 2x

Again, in ∆ABD, applying Pythagoras theorem, we have

(53√53)^2 = BC^2 + AC^2 => (53√53)^2 = x^2 + (2x)^2

We can get the unique value of x. Sufficient.

(2) \(ABD\) is an isosceles triangle.

Insufficient as we don't have values.

The correct answer: A

Hope this helps!

-Jay
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