If x and y are the lengths of the legs of a right triangle,

AbeNeedsAnswers wrote: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.

B

Source: Official Guide

Given: x and y are the lengths of the legs of a right triangle
We have something like this:


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

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

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

Cheers,
Brent

Hi All,

We're told that X and Y are the lengths of the legs of a RIGHT triangle. We're asked for the value of (X)(Y). This question can be solved with a mix of Geometry rules and TESTing VALUES.

(1) The hypotenuse of the triangle is 10√2.

With the information in Fact 1 - and the Pythagorean Theorem - we can create the following equation:

X^2 + Y^2 = (10√2)^2
X^2 + Y^2 = 200

Since this one equation has two variables, it's likely that there are LOTS of different possible values for X and Y - and as a result, there are probably lots of different values for (X)(Y). Here are two examples:

IF....
X=10 and Y= 10, then the answer to the question is (10)(10) = 100
X = 9 and Y = √119, then the answer to the question is 9√119
Fact 1 is INSUFFICIENT

(2) The area of the triangular region is 50.

With the information in Fact 2 - and the area formula - we can create the following equation:

Area = (1/2)(Base)(Height)
50 = (1/2)(X)(Y)
100 = (X)(Y)
This is the exact answer to the question that is asked. Since there's only one possible answer here, there's no more work needed.
Fact 2 is SUFFICIENT

Final Answer: B

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Rich