Official Guide Explanation:
Problem Solving #135

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Solution and Metadata

Question: 135
Page: 79
Difficulty: 5 (Moderate)
Category 1: Geometry > Quadrilaterals >

Explanation: Given that MPQT is a square, we know that all the sides of MPQT are equal. Call each side s. One of those sides is the width of rectangle MPRS, which was an area of 540. The width is s, and the length is s + 12, since PQ = s and QR = 12. Thus, we can use the area formula:

a = lw = (s + 12)(s)

540 = s² + 12s

s² + 12s - 540 = 0

Factor the quadratic:

(s + 30)(s - 18) = 0

s must equal 18 or -30. Since we're dealing with a physical shape, we can exclude the negative possibility; s = 18.

We're looking for the area of TQRS. We already know that QR is 12; now we know that RS (equal to PM, a side of the square) is 18, so the area is 12(18) = 216, choice (B).

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