2014 AMC 10A Problems/Problem 9
Contents
Problem
The two legs of a right triangle, which are altitudes, have lengths and . How long is the third altitude of the triangle?
Solution
We find that the area of the triangle is . By the Pythagorean Theorem, we have that the length of the hypotenuse is . Dropping an altitude from the right angle to the hypotenuse, we can calculate the area in another way.
Let be the third height of the triangle. We have
Solution 2
By the Pythagorean Theorem, we have that the length of the hypotenuse is . Notice that we now have a 30-60-90 triangle, with the angle between sides and equal to . Dropping an altitude from the right angle to the hypotenuse, we see that our desired height is (We can also check from the other side).
Video Solution (CREATIVE THINKING)
https://youtu.be/ZvoMN5tJHBk
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Video Solution
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See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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 | ||
All AMC 10 Problems and Solutions |
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