2014 AMC 10A Problems/Problem 22
Contents
Problem
In rectangle , and . Let be a point on such that . What is ?
Solution 1 (Trigonometry)
Note that . (It is important to memorize the sin, cos, and tan values of and .) Therefore, we have . Since is a triangle,
Solution 2 (No Trigonometry)
Let be a point on line such that points and are distinct and that . By the angle bisector theorem, . Since is a right triangle, and . Additionally, Now, substituting in the obtained values, we get and . Substituting the first equation into the second yields , so . Because is a triangle, .
~edited by ripkobe_745
Solution 3 Quick Construction (No Trigonometry)
Reflect over line segment . Let the point be the point where the right angle is of our newly reflected triangle. By subtracting to find , we see that is a right triangle. By using complementary angles once more, we can see that is a angle, and we've found that is a right triangle. From here, we can use the properties of a right triangle to see that
Solution 4 (No Trigonometry)
Let be a point on such that . Then Since , is isosceles.
Let . Since is , we have
Since is isosceles, we have . Since , we have Thus and .
Finally, by the Pythagorean Theorem, we have
~ Solution by Nafer
~ Edited by TheBeast5520
Note from williamgolly: When you find DE, note how ADE is congruent to a 30-60-90 triangle and you can easily find AE from there
Solution 5
First, divide all side lengths by to make things easier. We’ll multiply our answer by at the end. Call side length . Using the Pythagorean Theorem, we can get side is .
The double angle identity for sine states that: So, We know . In triangle , and . Substituting these in, we get our equation: which simplifies to
Now, using the quadratic formula to solve for . Because the length must be close to one, the value of will be . We can now find = and use it to find . . To find , we can use the Pythagorean Theorem with sides and , OR we can notice that, based on the two side lengths we know, is a triangle. So .
Finally, we must multiply our answer by , . .
~AWCHEN01
Solution 6 (Pure Euclidian Geometry)
We are going to use pure Euclidian geometry to prove .
Reflect rectangle along line . Let the square be as shown. Construct equilateral triangle .
Because , , and , by .
So, , .
Because , , , .
by .
So, . By the reflection, .
This solution is inspired by AoPS "Introduction to Geometry" page 226 problem 8.22, and page 433 problem 16.42.
~isabelchen
Solution 7 (Pure Euclidian Geometry)
We are going to use pure Euclidian geometry to prove .
Construct equilateral triangle , and let be the height of .
, , , .
by .
, , , by .
So, .
, , , , .
by .
So,
~isabelchen
Video Solution by Richard Rusczyk
https://www.youtube.com/watch?v=-GBvCLSfTuo
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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All AMC 10 Problems and Solutions |
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