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Right triangle DEF has angles in the ratio 1:2:3 and is . .

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by VJesus12 » Thu Nov 30, 2017 9:39 am
Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a circle with center C such that each vertex of DEF lies on the circumference of C. If the shortest side of DEF has length x, which of the following represents the area of circle C?

A. 2Ï€x

B. πx^2

C. sq. root 3Ï€x^2

D. 2Ï€x^2

E. 4 πx^2

The OA is B .

I don't know how can I solve this PS question. Experts, can you give me some help please? Thanks in advance.
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by [email protected] » Thu Nov 30, 2017 10:52 am
Hi VJesus12,

We're told that Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a circle with center C such that each vertex of DEF lies on the circumference of C. We're also told that the shortest side of DEF has length X. We're asked which of the following represents the area of circle C. This question is based on a few Geometry rules.

First, the only type of triangle that has angles in the ratio of 1:2:3 is a 30/60/90 right triangle.
Second, for a right triangle to be inscribed in a circle as described, the HYPOTENUSE of the right triangle will be the DIAMETER of the circle.
The ratio of the short side to the long side of a 30/60/90 is 1:2, so the hypotenuse of the right triangle triangle is equal to 2 radii (which is also the length of the short side - X).
Thus, the area of the circle is (pi)(X^2).

Final Answer: B

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