5.2.1 Indirect Proof and Inequalities in One Triangle (continued)
Example 4 Finding Side Lengths When the lengths of two sides of a triangle are set, the third side cannot be any length. The third side can be a range of values that is limited by its relationship to the lengths of the other two sides.
A triangle can be formed on this map where the three cities, Austin, Mason, and San Antonio, are the triangle’s vertices. Since the distance from Austin to Mason and the distance from Mason to San Antonio are known, the lengths of two of the triangle’s sides are known. Let d represent the distance from Austin to San Antonio. Use the two known lengths and d to write the inequalities. However, note that the inequality d + 111 > 108 can be ignored because that solution set, d > − 3, includes negative values and distance can never be negative. To find the range of lengths for the third side of a triangle when two side lengths are known, first assign a variable for the length of the third side. Here, the length of the unknown side is represented by s . Then write the three inequalities where each relates the sum of two sides to the other side. Here, those three inequalities are s + 5 > 9, s + 9 > 5, and 5 + 9 > s . Solve each inequality. s > 4 s > − 4 14 > s , or s < 14 Now find the range of values that satisfies all three inequalities. The range between 4 and 14 satisfies all 3 inequalities. Therefore, the range of side lengths for this triangle is 4 < s < 14.
Example 5 Travel Application
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