What is G?

Geometry Level 2

$ABCD$ is a square of side length 10, and a circle is inscribed within.
$E$ and $F$ are midpoints of $AB$ and $CD$ respectively.
$EC$ intersects the circle again at $G$ .

What is the length of $FG$ ?

3 \sqrt{2}

5

2 \sqrt{5}

4

1 solution

Shaun Leong
Oct 24, 2016

$\angle EGF = 90^\circ = \angle FGC$ $\angle EFG = 90^\circ - \angle GFC = \angle FCG$ Since all 3 corresponding angles are equal, $\triangle EGF \sim \triangle FGC$ .

They are also similar to $\triangle EFC$ and are $1:2:\sqrt5$ triangles.

$FG = 2CG$ $=\frac{2}{5} EC$ $=\frac{2}{5} \times 5\sqrt5$ $=\boxed{2\sqrt5}$

Nice recognition of the similar triangles.

When I set up the problem, I was using the (slightly hidden) fact that $FG$ is the altitude to $EC$ , and so we can find the height through the area.

Calvin Lin Staff - 4 years, 7 months ago

What is G?

1 solution

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