2004 AMC 12A Problems/Problem 18
From AoPSWiki
(Redirected from 2004 AMC 10A Problems/Problem 22)
- The following problem is from both the 2004 AMC 12A #18 and 2004 AMC 10A #22, so both problems redirect to this page.
Problem
Square 
has side length 
. A semicircle with diameter 
is constructed inside the square, and the tangent to the semicircle from 
intersects side 
at 
. What is the length of 
?
Contents |
Solution
Solution 1
Let the point of tangency be 
. By the Two Tangent Theorem 
and 
. Thus 
. The Pythagorean Theorem on 
yields
Hence 
.
Solution 2
Clearly, 
. Thus, the sides of right triangle 
are in arithmetic progression. Thus it is similar to the triangle 
and since 
, 
.
See also
| 2004 AMC 12A (Problems • Resources) | ||
| Preceded by Problem 17 | Followed by Problem 19 | |
| 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 | ||
| 2004 AMC 10A (Problems • Resources) | ||
| Preceded by Problem 21 | Followed by Problem 23 | |
| 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 | ||
