The base of a triangle is 6 cm shorter than the altitude to that base. How do you find the length of the base and the altitude if the are of the triangle is 56 cm^2?

1 Answer
Zor Shekhtman
Feb 8, 2016

Base #a=8#
Altitude #h=14#

Explanation:

Let's approach this problem algebraically.

Assume that the triangle's altitude is #h# and a corresponding base is #a#.
Then from the relationship between their lengths we can derive the following equation:
#a = h-6#
Since the area is given, the second equation is
#1/2ah = 56#

All we have to do now is to solve this system of two equations with two unknowns.
Use an expression for #a# from the first equation and substitute it into the second:
#1/2(h-6)h=56#

This is a quadratic equation, which in a more traditional form looks like
#h^2-6h-112 = 0#

Its solutions are
#h_(1,2) = (6+-sqrt(36+448))/2 = (6+-22)/2=3+-11#

Negative solution should be discarded since we are dealing with lengths, so the only solution for #h# is #14#.
Then the base #a# is #8#.

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