2008 iTest Problems/Problem 62

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Problem

Find the number of values of $x$ such that the number of square units in the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer.

Solution

Let $\theta$ be the angle between the two sides with length $65$. The area of the triangle is equal to $\tfrac12 \cdot 65^2 \cdot \sin(\theta) = \tfrac{4225}{2}\sin(\theta)$.


Note that $0^\circ < \theta < 180^\circ$, so we know that $0 < \sin(\theta) \le 1$. Since $\tfrac{4225}{2} = 2112.5$ is not an integer, for every integer from $1$ to $2112$, there are two possible values of $\theta$ since $\theta$ can either be acute or obtuse. Each $\theta$ corresponds to exactly one value of $x,$ so there are $2112 \cdot 2 = \boxed{4224}$ values of $x$ such that the area of the triangle is a positive integer.

See Also

2008 iTest (Problems)
Preceded by:
Problem 61
Followed by:
Problem 63
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