Big Ideas Math Geometry, 2014
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Big Ideas Math Geometry, 2014 View details
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Exercise 4 Page 523

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

Do the Segment Lengths Form a Triangle? Yes.
Is the Triangle Acute, Right, or Obtuse? Right triangle.

Practice makes perfect

We have been given the following segment lengths. We will investigate whether these segment lengths form a triangle. Then, we will determine whether the triangle is acute, right, or obtuse.

16, 30, and 34

Let's start!

Step I

In this step, we will use the Triangle Inequality Theorem to verify that the segment lengths form a triangle.


Inequality Check
16+30 ? > 34
46 > 36 âś“
16 + 34 ? > 30 50 > 30 âś“
34 + 30 ? > 16 64 > 16 âś“

We can see that the segments with lengths of 16, 30, and 34 satisfy the Triangle Inequality Theorem. Therefore, they form a triangle.

Step II

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 16, 30 and 34. Since 34 is the greatest of the numbers, we will let c be 34. We will also arbitrarily let a be 16 and b be 30. 16^2+30^2 ? 34^2 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.
16^2+30^2 ? 34^2
256 + 900 ? 1156
1156 = 1156
Referring back to our table, we can conclude that the side lengths 16, 30, and 34 form a right triangle.