Circles And Triangles

Geometry Level 2

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 the triangle $FGH$ ?

300 270 240 50

2 solutions

A Former Brilliant Member
Apr 4, 2018

Since FD is the diameter, it is also the hypotenuse of right triangle FED. By pythagorean theorem, FE=12. By similar triangles,

FH/15 = 12/5

FH = 36

Since FED is a right triangle, GFH is also a right triangle, so the area is

A = 0.5 * 36 * 15 = 270

Rajdeep Ghosh
Aug 27, 2016

Let’s begin by focusing on triangle FED. The angle ∠E spans a diameter, so ∠E = 90°. Thus, triangle FED is a right triangle with hypotenuse FD = 13 and leg ED = 5. It will save you a tremendous amount of calculations here if you already have memorized the 5-12-13 Pythagorean Triplet. Thus, FE = 12. Area = (0.5)bh = (0.5)(12)(5) = 30.

Because ED and GH are parallel, all the angles are equal, and the two triangles are similar. From ED = 5 to GH = 15 we scale up by a scale factor of k = 3. Lengths are multiplied by the scale factor, and areas are multiplied by the scale factor squared, k^2 = 9. $30\times 9 = 270$ is the area of FGH.

Circles And Triangles

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