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Triangle Inequalities

- 4.6
- Apply inequalities in one triangle.

The positions of the longest and shortest sides

of a triangle are related to the positions of the

largest and smallest angles.

Example 1 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.

Example 2 Ordering Triangle Side Lengths and

Angle Measures

Write the sides in order from shortest to longest.

m?R 180 (60 72) 48

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 3 Applying the Triangle Inequality

Theorem

Tell whether a triangle can have sides with the

given lengths. Explain.

8, 13, 21

7, 10, 19

Noby the Triangle Inequality Theorem, a triangle

cannot have these side lengths.

Example 4 Applying the Triangle Inequality

Theorem

Tell whether a triangle can have sides with the

given lengths. Explain.

2.3, 3.1, 4.6

?

?

?

Yesthe sum of each pair of lengths is greater

than the third length.

Example 5 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

n2 1

3n

4 6

(4)2 1

3(4)

10

15

12

Example 5 Continued

Step 2 Compare the lengths.

?

?

?

Yesthe sum of each pair of lengths is greater

than the third length.

Example 6 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.

x 8 gt 13

x 13 gt 8

8 13 gt x

x gt 5

x gt 5

21 gt x

Combine the inequalities. So 5 lt x lt 21. The

length of the third side is greater than 5 inches

and less than 21 inches.

Example 5 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 gt 51

x 51 gt 46

46 51 gt x

? Inequal. Thm.

x gt 5

x gt 5

97 gt x

Subtr. Prop. of Inequal.

5 lt x lt 97

Combine the inequalities.

The distance from San Francisco to Oakland is

greater than 5 miles and less than 97 miles.

Check It Out! Example 5

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 gt 50

x 50 gt 22

22 50 gt x

? Inequal. Thm.

x gt 28

x gt 28

72 gt x

Subtr. Prop. of Inequal.

28 lt x lt 72

Combine the inequalities.

The distance from Seguin to Johnson City is

greater than 28 miles and less than 72 miles.

Homework

- Pg 216 1-20
- Extra help and problems are on the next slides.

Go to my website to see them.

Lesson Quiz Part I

1. 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.

5 cm lt x lt 29 cm

No 2.7 3.5 is not greater than 9.8.

5. Ray wants to place a chair so it is 10 ft from

his television set. Can the other two

distances shown be 8 ft and 6 ft? Explain.

Yes the sum of any two lengths is greater than

the third length.

Check It Out! Example 1

Write the angles in order from smallest to

largest.

The angles from smallest to largest are ?B, ?A,

and ?C.

Check It Out! Example 2

Write the sides in order from shortest to longest.

m?E 180 (90 22) 68

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.

x 22 gt 17

x 17 gt 22

22 17 gt x

x gt 5

x gt 5

39 gt x

Combine the inequalities. So 5 lt x lt 39. The

length of the third side is greater than 5 inches

and less than 39 inches.

Example

- A landscape architect is designing a triangular

deck. She wants to place benches in the two

largest corners. In which corners should she

place the benches?

27ft

?

21ft

19ft

Homework

- P. 216 1-24

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