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If x and y are the lengths of the legs of a right triangle, what is the value of xy ?

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If x and y are the lengths of the legs of a right triangle, what is the value of xy ?

(1) The hypotenuse of the triangle is 10√2.
(2) The area of the triangular region is 50.

Answer: B
Source: Official Guide

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BTGModeratorVI wrote:
Wed Mar 25, 2020 6:33 am
If x and y are the lengths of the legs of a right triangle, what is the value of xy ?

(1) The hypotenuse of the triangle is 10√2.
(2) The area of the triangular region is 50.

Answer: B
Source: Official Guide
Given: x and y are the lengths of the legs of a right triangle
We have something like this:
Image

Target question: What is the value of xy?

Statement 1: The hypotenuse of the triangle is [m]10[square_root]2[/square_root][/m].
There are infinitely-many different right triangles that meet this condition. Here are two:
Case a: x = 10 and y = 10
Image
CHECK: If h = the hypotenuse, then we get 10² + 10² = h²
Solve: 200 = h²
So, h = √200 = 10√2
In this case, the answer to the target question is xy = (10)(10) = 100

Case b: x = √50 and y = √150
Image
CHECK: If h = the hypotenuse, then we get (√50)² + (√150)² = h²
Solve: 200 = h²
So, h = √200 = 10√2
In this case, the answer to the target question is xy = (√50)(√150) = √7500 = 50√3

Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT


Statement 2: The area of the triangular region is 50
Area of triangle = (base)(height)/2
So, we can write: (x)(y)/2 = 50
Multiply both sides by 2 to get: xy = 100
So, the answer to the target question is xy = 100
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

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Legs of a right angle triangle = two sides of the right angle triangle other than the hypotenuse. i.e legs = base and height
Question=> What is the value of xy?
Statement 1: The hypotenuse of the triangle is 10√2.
From Pythagoras theorem,
$$\left(10\cdot\sqrt{2}\right)^2=x^2+y^2$$
$$\left(10^2\cdot2\right)=x^2+y^2$$
$$\left(100\cdot2\right)=x^2+y^2$$
$$200=x^2+y^2$$
The value of x and y remains unknown, hence, statement 1 is NOT SUFFICIENT.

Statement 2: the area of the triangular region is 50.
Area of traingle = 1/2 * base * height
$$50=\frac{1}{2}\cdot x\cdot y$$
$$xy=100$$
Statement 2 is SUFFICIENT.

Since only statement 2 is sufficient, then option B is correct.