2005 AMC 10A Problems/Problem 23
From AoPSWiki
Problem
Let 
be a diameter of a circle and let 
be a point on with 
. Let 
and 
be points on the circle such that 
and 
is a second diameter. What is the ratio of the area of 
to the area of 
?
. The area of
Solution
http://img443.imageshack.us/img443/8034/circlenc1.png
is 
of diameter and 
is 
- = 
.
is the radius of the circle, so using the Pythagorean theorem height 
of 
is 
= 
. This is also the height of the .
Area of the 
is 
= 
.
The height of can be found using the area of and 
as base.
Hence the height of is 
= 
.
The diameter is the base for both the triangles and .
Hence, the ratio of the area of to the area of is
= 
See also
| 2005 AMC 10A (Problems • Resources) | ||
| Preceded by Problem 22 | Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
