2011 AMC 10B Problems/Problem 18
Contents
Problem
Rectangle has and . Point is chosen on side so that . What is the degree measure of ?
Solution 1
It is given that . Since and are alternate interior angles and , . Use the Base Angle Theorem to show . We know that is a rectangle, so it follows that . We notice that is a triangle, and . If we let be the measure of then
Solution 2
Let . If we let , we have that , by the Pythagorean Theorem, and similarily, . Applying LOC, we see that and YAY!!! We have two equations for two variables... that are terribly ugly. Well, we'll try to solve it. First of all, note that , so solving for in terms of , we get that . The equation now becomes
Simplifying, we get
Now, we apply the quartic formula to get
We can easily see that is an invalid solution. Thus, .
Finally, since , , where is any integer. Converting to degrees, we have that . Since , we have that .
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See Also
2011 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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All AMC 10 Problems and Solutions |
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