In the diagram above, \(ED\) is parallel to \(GH,\) and the circle has a diameter of 13. If \(ED = 5\) and \(GH = 15,\)

[Vincen]

In the diagram above, \(ED\) is parallel to \(GH,\) and the circle has a diameter of 13. If \(ED = 5\) and \(GH = 15,\) what is the area of triangle \(FGH?\)

(A) 240
(B) 270
(C) 300
(D) 330
(E) 360

[spoiler]OA=B[/spoiler]

Source: Magoosh

[Scott@TargetTestPrep]

gpp_img4.png

In the diagram above, \(ED\) is parallel to \(GH,\) and the circle has a diameter of 13. If \(ED = 5\) and \(GH = 15,\) what is the area of triangle \(FGH?\)

(A) 240
(B) 270
(C) 300
(D) 330
(E) 360

[spoiler]OA=B[/spoiler]

Source: Magoosh

We see that triangle FGH and triangle DEF are similar triangles, and since GH (a side of triangle GHF) is 3 times ED (the corresponding side of triangle DEF), the area of triangle FGH will be 3^2 = 9 times the area of triangle DEF. Thus, if we can determine the area of triangle DEF, then we can determine the area of triangle FGH. So let’s determine the area of triangle DEF.
Triangle DEF is inscribed in circle Q, whose diameter is 13. Since FD is a diameter of circle Q, triangle DEF must be a right triangle. Since we also know ED = 5, triangle DEF must be a 5-12-13 right triangle. That is, EF = 12, and hence, the area of triangle DEF is ½ x 12 x 5 = 30. Finally, since the area of triangle FGH is 9 times the area of triangle DEF, the area of triangle FGH is 9 x 30 = 270.

Answer: B