In a semicircle, an inscribed angle that intercepts the diameter is 90 degrees. Thus, ∠XYZ is a right angle, making triangle XYZ a right triangle. The line drawn through the right angle at Y is a height of triangle XYZ because it is perpendicular to line XZ (which we can consider the base).
In any right triangle, a height drawn through the right angle to the opposite side creates 3 similar triangles.
Thus, the small right triangle at the bottom, the larger right triangle at the top, and right triangle XYZ are all similar.
The corresponding sides of similar triangles must yield the same proportion.
In the small right triangle at the bottom, the shorter leg = s, the longer leg = 4.
In the larger right triangle at the top, the shorter leg = 4, the longer leg = r.
Since shorter leg:longer leg must be the same for each triangle, we get:
s/4 = 4/r.
Statement 1:
If r=8, then s/4 = 4/8, and s=2.
Diameter = r+s = 8+2 = 10.
Sufficient.
Statement 2:
If s=2, then 2/4 = 4/r, and r=8.
Diameter = r+s = 8+2 = 10.
Sufficient.
The correct answer is
D.
Please confirm the OA. C is not the correct answer.
For another discussion of the 3 similar triangles formed when a height is drawn through the right angle of a right triangle, please check the following thread:
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