2016 AMC 12B Problems/Problem 17
Problem
In shown in the figure, , , , and is an altitude. Points and lie on sides and , respectively, so that and are angle bisectors, intersecting at and , respectively. What is ?
Solution
Get the area of the triangle by heron's formula: Use the area to find the height AH with known base BC: Apply angle bisector theorem on triangle and triangle , we get and , respectively. From now, you can simply use the answer choices because only choice has in it and we know that the segments on it all have integral lengths so that will remain there. However, by scaling up the length ratio: and . we get .
Solution 2: (choices+faster) As in the above solution, let's find the area. By heron's, it is indeed . Because is an altitude and we are given a base, let's find . We set up the area is equal to and since the answer should have in it so our answer is .
See Also
2016 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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All AMC 12 Problems and Solutions |
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