Difference between revisions of "2014 AMC 10A Problems/Problem 9"
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==Solution 3 (Simple)== | ==Solution 3 (Simple)== | ||
− | From solution 1, we know the 3rd altitude of a right triangle is the product of the 2 legs that are not the hypotenuse, divided by the length of the hypotenuse. The product of the 2 legs of this triangle is <math>12\sqrt{3}</math>. By the [[Pythagorean Theorem]], the length of the hypotenuse is <math>\sqrt{(2\sqrt{3})^2+6^2}=4\sqrt{3}</math>. | + | From solution 1, we know the 3rd altitude of a right triangle is the product of the 2 legs that are not the hypotenuse, divided by the length of the hypotenuse. The product of the 2 legs of this triangle is <math>12\sqrt{3}</math>. By the [[Pythagorean Theorem]], the length of the hypotenuse is <math>\sqrt{(2\sqrt{3})^2+6^2}=4\sqrt{3}</math>. The length of the third altitude is <math>\frac{12\sqrt{3}}{4\sqrt{3}}=\boxed{\textbf{(C)}\ 3}</math> |
− | ~shunyipanda | + | ~[[shunyipanda]] |
==Video Solution (CREATIVE THINKING)== | ==Video Solution (CREATIVE THINKING)== | ||
https://youtu.be/ZvoMN5tJHBk | https://youtu.be/ZvoMN5tJHBk |
Latest revision as of 17:49, 22 October 2025
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 1
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
Note: The third altitude of a right triangle is always the product of the lengths of the two legs divided by the hypotenuse.
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).
Solution 3 (Simple)
From solution 1, we know the 3rd altitude of a right triangle is the product of the 2 legs that are not the hypotenuse, divided by the length of the hypotenuse. The product of the 2 legs of this triangle is . By the Pythagorean Theorem, the length of the hypotenuse is
. The length of the third altitude is
~shunyipanda
Video Solution (CREATIVE THINKING)
https://youtu.be/ZvoMN5tJHBk
~Education, the Study of Everything
Video Solution
~savannahsolver
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 |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.