McGraw Hill Glencoe Geometry, 2012
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McGraw Hill Glencoe Geometry, 2012 View details
2. The Pythagorean Theorem and Its Converse
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Exercise 21 Page 553

Compare the square of the largest side length to the sum of the squares of the other two side lengths.

Is a triangle?: Yes
Classification: Obtuse
Explanation: See solution.

Practice makes perfect

We want to determine whether the given side lengths can be the measures of a triangle. To do so, we will use the following theorem.

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Let's check if that is the case.
7+15 ? > 21 21+7 ? > 15 15+21 ? > 7
22 > 21 28 > 15 36 > 7

Therefore, these sides lengths can indeed form a triangle. Now, 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, 15, and 21. Since 21 is the greatest of the numbers, we will let c be 21. We will also arbitrarily let a be 7 and b be 15. 7^2+15^2 ? 21^2 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.
7^2+15^2 ? 21^2
49+225 ? 441
274 < 441
Referring back to our table, we can conclude that the side lengths 7, 15, and 21 form an obtuse triangle.