Apply inequalities in one triangle.

Objectives

Triangle inequality theorem

Vocabulary

The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles.

Example 1a: Ordering Triangle Side Lengths and Angle Measures

Write the angles in order from smallest to largest.

The angles from smallest to largest are F, H and G.

The shortest side is , so the smallest angle is F.

The longest side is , so the largest angle is G.

Example 2B: Ordering Triangle Side Lengths and Angle Measures

Write the sides in order from shortest to longest.

mR = 180° – (60° + 72°) = 48°

The smallest angle is R, so the shortest side is .

The largest angle is Q, so the longest side is .

The sides from shortest to longest are

Check It Out! Example 1a

Write the angles in order from smallest to largest.

The angles from smallest to largest are B, A, and C.

The shortest side is , so the smallest angle is B.

The longest side is , so the largest angle is C.

Check It Out! Example 2b

Write the sides in order from shortest to longest.

mE = 180° – (90° + 22°) = 68°

The smallest angle is D, so the shortest side is .

The largest angle is F, so the longest side is .

The sides from shortest to longest are

A triangle is formed by three segments, but not every set of three segments can form a triangle.

A certain relationship must exist among the lengths of three segments in order for them to form a triangle.

Example 2A: Applying the Triangle Inequality Theorem

Tell whether a triangle can have sides with the given lengths. Explain.

7, 10, 19

No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

Example 2B: Applying the Triangle Inequality Theorem

Tell whether a triangle can have sides with the given lengths. Explain.

2.3, 3.1, 4.6

Yes—the sum of each pair of lengths is greater than the third length.

Example 2C: Applying the Triangle Inequality Theorem

Tell whether a triangle can have sides with the given lengths. Explain.

n + 6, n2 – 1, 3n, when n = 4.

Step 1 Evaluate each expression when n = 4.

n + 6

4 + 6

10

n2 – 1

(4)2 – 1

15

3n

3(4)

12

Example 2C Continued

Step 2 Compare the lengths.

Yes—the sum of each pair of lengths is greater than the third length.

Check It Out! Example 2a

Tell whether a triangle can have sides with the given lengths. Explain.

8, 13, 21

No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

Check It Out! Example 2b

Tell whether a triangle can have sides with the given lengths. Explain.

6.2, 7, 9

Yes—the sum of each pair of lengths is greater than the third side.

Check It Out! Example 2c

Tell whether a triangle can have sides with the given lengths. Explain.

t – 2, 4t, t2 + 1, when t = 4

Step 1 Evaluate each expression when t = 4.

t – 2

4 – 2

2

t2 + 1

(4)2 + 1

17

4t

4(4)

16

Check It Out! Example 2c Continued

Step 2 Compare the lengths.

Yes—the sum of each pair of lengths is greater than the third length.

Example 3: Finding Side Lengths

The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side.

Let x represent the length of the third side. Then apply the Triangle Inequality Theorem.

Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches.

x + 8 > 13

x > 5

x + 13 > 8

x > –5

8 + 13 > x

21 > x

Check It Out! Example 3

The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side.

Let x represent the length of the third side. Then apply the Triangle Inequality Theorem.

Combine the inequalities. So 5 < x < 39. The length of the third side is greater than 5 inches and less than 39 inches.

x + 22 > 17

x > –5

x + 17 > 22

x > 5

22 + 17 > x

39 > x

Example 4: Travel Application The figure shows the approximate distances between cities in California. What is the range of distances from San Francisco to Oakland?

Let x be the distance from San Francisco to Oakland.

x + 46 > 51

x > 5

x + 51 > 46

x > –5

46 + 51 > x

97 > x

5 < x < 97 Combine the inequalities.

Δ Inequal. Thm.

Subtr. Prop. of Inequal.

The distance from San Francisco to Oakland is greater than 5 miles and less than 97 miles.

Check It Out! Example 4

The distance from San Marcos to Johnson City is 50 miles, and the distance from Seguin to San Marcos is 22 miles. What is the range of distances from Seguin to Johnson City?

Let x be the distance from Seguin to Johnson City.

x + 22 > 50

x > 28

x + 50 > 22

x > –28

22 + 50 > x

72 > x

28 < x < 72 Combine the inequalities.

Δ Inequal. Thm.

Subtr. Prop. of Inequal.

The distance from Seguin to Johnson City is greater than 28 miles and less than 72 miles.

Lesson Quiz: Part I1. Write the angles in order from smallest to

largest.

2. Write the sides in order from shortest to longest.

C, B, A

Lesson Quiz: Part II

3. The lengths of two sides of a triangle are 17 cm and 12 cm. Find the range of possible lengths for the third side.

4. Tell whether a triangle can have sides with lengths 2.7, 3.5, and 9.8. Explain.

No; 2.7 + 3.5 is not greater than 9.8.

5 cm < x < 29 cm

5. Ray wants to place a chair so it is 10 ft from his television set. Can the other two distancesshown be 8 ft and 6 ft? Explain.

Yes; the sum of any two lengths is greater than the third length.