Question 30

The base of a triangle is $$4$$ units less than the altitude drawn to it. The area of the triangle is $$96\text{ units}^2$$. The ratio of the base to the height is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are relatively prime. Then the value of $$p+q$$ is

Correct Answer: 7

Solution

Let the height be $$h$$, so the base is $$h-4$$. The area condition gives $$\frac{1}{2}h(h-4)=96$$, or $$h^2-4h-192=0$$. The positive root is $$h=16$$, so the base is $$12$$. Therefore the ratio is $$\frac{12}{16}=\frac{3}{4}$$ and $$p+q=3+4=7$$.

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