Triangle PQR is a right triangle. QS is a height drawn through the right angle to the opposite side.
In any right triangle, a height drawn through the right angle to the opposite side creates 3 similar triangles.
We can prove this by plugging in for the angles measurements. If ∠RPQ = 20, then ∠PQS = 70, since the sum of the 2 angles must be 90. This forces ∠RQS to be 20. This forces ∠QRS to be 70. The result is that all 3 triangles (PQS, RQS, and PQR) have the same combination of angles: 20-70-90. Thus, the 3 triangles are similar.
The corresponding sides of similar triangles must yield the same proportion.
In triangle PQS, the shorter leg = QS, the longer leg = PS = 16.
In triangle RQS, the shorter leg = RS = 9, the longer leg = QS.
Since shorter leg:longer leg must be the same for each triangle, we get:
QS/16 = 9/QS
(QS)² = 144
QS = 12.
Thus, in PQR, h = 12 and b = PS + SR = 16+9 = 25.
Area = 1/2*b*h = 1/2(25)(12) = 150.
The correct answer is
D.
Another approach:
In triangle PQR, b = PS + SR = 16+9 = 25.
Now we can plug in the answer choices, which represent the area of triangle PQR. Since b=25, the correct answer likely will be divisible by 25. Let's start with answer choice D.
Answer choice D: A = 150
1/2(25)(h) = 150
h = 12.
Now let's see if h=12 makes all the geometry work.
The legs of triangle PQS are 12 and 16, yielding a multiple of a 3:4:5 triangle. 3:4:5 = 12:16:20. Thus, PQ=20.
The legs of triangle RQS are 9 and 12, yielding another multiple of a 3:4:5 triangle. 3:4:5 = 9:12:15. Thus, QR = 15.
The legs of triangle PQR are 15 and 20 and the hypotenuse is 25, yielding another multiple of a 3:4:5 triangle. 3:4:5 = 15:20:25.
Since all the geometry works, success!
The correct answer is
D.
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