Problem:
The line 12x+5y=60 forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
Answer Choices:
A. 20
B. 17360​
C. 5107​
D. 243​
E. 13281​
Solution:
Label the vertices of the triangle A=(0,0),B=(5,0), and C=(0,12). By the Pythagorean Theorem BC=13. Two altitudes are 5 and 12. Let AD be the third altitude. The area of this triangle is 30, so 21​⋅AD⋅BC=30. Therefore AD=BC2⋅30​=1360​. The sum of the lengths of the altitudes is 5+12+1360​=(E)13281​​.
The problems on this page are the property of the MAA's American Mathematics Competitions
