Question

A triangle has an area of 88 square inches. Find the length of the base if the base is 5 inches more than the height.

Answer 1

The base is 16 inches.

The formula for the area of a triangle is

A = 1/2bh

We know that b = h+5, so we can rewrite this as:

A = 1/2(h+5)(h)

Using the area that we have, we now have:

88 = 1/2(h+5)(h)

We can cancel the 1/2 by multiplying by 2:
88*2 = (1/2)(h+5)(h)*2
176 = (h+5)(h)

Using the distributive property, we have:
176 = h² + 5h

We want quadratic equations to be set equal to 0, so we will subtract 176 from both sides:
176 - 176 = h² + 5h - 176
0 = h² + 5h - 176

Using the quadratic formula:
$h=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ =\frac{-5\pm \sqrt{5^2-4(1)(-176)}}{2(1)}=\frac{-5\pm \sqrt{25--704}}{2} \\ \\=\frac{-5\pm \sqrt{25+704}}{2}=\frac{-5\pm \sqrt{729}}{2} \\ \\=\frac{-5\pm 27}{2}=\frac{-5-27}{2}\text{ or }\frac{-5+27}{2} \\ \\=\frac{-32}{2}\text{ or }\frac{22}{2}=-16\text{ or }11$

Since a negative length makes no sense, we know that h=11.

The base is 5 inches longer than the height, so b = 11+5 = 16.