Exercise 7 Page 514

We want to determine whether the triangle formed by the given side lengths is acute, right, or obtuse. To do so, we will compare the square of the largest side length to the sum of the squares of the other two side lengths. Let $a,$ $b,$ and $c$ be the lengths of the sides, with $c$ being the longest.

Condition	Type of Triangle
$a^2+b^2<c^2$	Obtuse triangle
$a^2+b^2=c^2$	Right triangle
$a^2+b^2>c^2$	Acute triangle

Let's now consider the given side lengths $7,$ $8,$ and $9.$ Since $9$ is the greatest of the numbers, we will let $c$ be $9.$ We will also arbitrarily let $a$ be $7$ and $b$ be $8.$ Let's simplify the above statement to determine whether the left-hand side is less than, equal to, or greater than the right-hand side. Referring back to our table, we can conclude that the side lengths $7,$ $8,$ and $9$ form an acute triangle.