2020 CIME I Problems/Problem 5
Problem 5
Let be a rectangle with sides
and let
be the reflection of
over
. If
and the area of
is
, find the area of
.
Solution
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Let be the center of rectangle
. Because
is the reflection of
over
and
degrees, we have
degrees. This means
lies on the circle with diameter
, or the circumcircle of rectangle
. We are given
, so by symmetry,
. Since the three lengths are equal and
degrees, we must have
\overarc{ED}=60
\triangle OBC
\triangle OCE
\triangle OED
ECBD
144
\triangle OBC
\frac{1}{3} \cdot 144 = 48
\frac{1}{4}
ABCD
48 \cdot 4 = \boxed{192}$.
2020 CIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All CIME Problems and Solutions |
The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions.