Question

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?

Answer 1

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`.