SOLUTION: the hight of a tirangle is 3 inches less than the lenght of the base. if the area is 54 inches squared. find the height of the base

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Question 370177: the hight of a tirangle is 3 inches less than the lenght of the base. if the area is 54 inches squared. find the height of the base
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = the length of the base. Then, since the height is 3 inches less, the height = x-3.

The formula for area of a triangle is:
A+=+%281%2F2%29%2Ab%2Ah
We know that the area is 54 square inches and we have expressions for the other two variables in the equation. So we can replace all three variables in the equation:
54+=+%281%2F2%29%2A%28x%29%2A%28x-3%29
We can solve this equation. First we simplify:
54+=+%281%2F2%29%2A%28x%5E2+-3x%29
54+=+%281%2F2%29%2Ax%5E2+-+%283%2F2%29%2Ax%29
As part of simplifying, I like to eliminate the fractions. This can be done by multiplying both sides of the equation by the Lowest Common Denominator (LCD) of all the denominators. The LCD here is easy since both denominators are the same. Multiplying each side by 2:
2%2A%2854%29+=+2%2A%28%281%2F2%29%2Ax%5E2+-+%283%2F2%29%2Ax%29%29
On the right side we use the Distributive Property:
2%2A%2854%29+=+2%2A%281%2F2%29%2Ax%5E2+-+2%2A%283%2F2%29%2Ax%29%29
which simplifies to:
108+=+x%5E2+-+3x%29%29
This is a quadratic equation so we want one side to be zero. Subtracting 108 from each side we get:
0+=+x%5E2+-3x+-108
Now we factor (or use the Quadratic Formula):
0 = (x - 12)(x + 9)
From the Zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
x - 12 = 0 or x + 9 = 0
Solving each of these we get:
x = 13 or x = -9

Since x is the length of the base and since the length of a base cannot be negative, we will reject the negative answer. So the base is 12 inches. And the height, which is x-3, is 9 inches.