# GMAT OFFICIAL GUIDE DS – If arc PQR above is a semicircle…

## Solution:

We are given a triangle inscribed inside a semicircle. When a triangle is inscribed inside a semicircle, the triangle must be a right triangle. Therefore, triangle PQR is a right triangle with a right angle at Q. With this knowledge, let’s start by sketching the diagram.

Notice that in sketching the diagram we added point S to create line segment QS. Thus, we see that a represents the length of side PS and b represents the length of side SR. We also added in that angle PQR is a 90-degree angle because it’s a right angle.
We see now that we have 3 right triangles.

1) Right triangle PQR

2) Right triangle SQR

3) Right triangle PQS

We can let side PQ = x and side QR = y.

Using the Pythagorean theorem, we can create a few equations.

Equation 1

a^2 + 2^2 = x^2

a^2 + 4 = x^2

Equation 2

b^2 + 2^2 = y^2

b^2 + 4 = y^2

Equation 3

x^2 + y^2  = (a + b)^2

x^2 + y^2  = a^2 + b^2 + 2ab

We can combine equations one and two by adding them together. We obtain:

a^2 + 4 + b^2 + 4  = x^2 + y^2

a^2 + b^2 + 8  =  x^2 + y^2

Since a^2 + b^2 + 8 = x^2 + y^2, we can substitute a^2 + b^2 + 8 for x^2 + y^2 in the equation x^2 + y^2  = a^2 + b^2 + 2ab.

a^2 + b^2 + 8 = a^2 + b^2 + 2ab

8 = 2ab

4 = ab

We are asked to determine the value of diameter PR, which is the sum of a and b. If we can determine a value for a or b, we will have enough information to determine the values for a and b, and thus determine a value for the sum of a and b.

Statement One Alone:

a = 4

Since a = 4 and ab = 4, we know that b = 1. Thus, the sum of a and b is 5. Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

b = 1

Since b = 1 and ab = 4, we know that a = 4. Thus, the sum of a and b is 5. Statement two alone is also sufficient to answer the question.