Warm-Up Exercises

1. An isosceles triangle has side lengths 20 meters, 26 meters, and 26 meters. Find the length of the altitude to the base.

122. Solve 18 = x.

ANSWER 24 m

ANSWER 36

Warm-Up Exercises

3. Evaluate (4 cos 10º)(10 sin 20º).

4. Evaluate .10tan 88º

ANSWER 13.47

0.3492ANSWER

Warm-Up ExercisesEXAMPLE 1 Find angle measures in a regular polygon

a. m AFB

In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure.

SOLUTION

AFB is a central angle, so m AFB = , or 72°.

360°

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Warm-Up ExercisesEXAMPLE 1 Find angle measures in a regular polygon

b. m AFG

In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure.

SOLUTION

FG is an apothem, which makes it an altitude of isosceles ∆AFB. So, FG bisects AFB and m

AFG = m AFB = 36°.12

Warm-Up ExercisesEXAMPLE 1 Find angle measures in a regular polygon

c. m GAF

In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure.

SOLUTION

The sum of the measures of right ∆GAF is 180°.So, 90° + 36° + m GAF = 180°, and m GAF = 54°.

Warm-Up ExercisesGUIDED PRACTICE for Example 1

In the diagram, WXYZ is a square inscribed in P.

1. Identify the center, a radius, an apothem, and a central

angle of the polygon.

P, PY or XP, PQ, XPY.

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Example 1

2. Find m XPY, m XPQ, and m PXQ.

90°, 45°, 45°

ANSWER

Warm-Up ExercisesEXAMPLE 2 Find the area of a regular polygon

DECORATING

You are decorating the top of a table by covering it with small ceramic tiles. The table top is a regular octagon with 15 inch sides and a radius of about 19.6 inches. What is the area you are covering?

SOLUTION

STEP 1 Find the perimeter P of the table top. An octagon has 8 sides, so P = 8(15) = 120 inches.

Warm-Up ExercisesEXAMPLE 2

STEP 2

So, QS = (QP) = (15) = 7.5 inches.12

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To find RS, use the Pythagorean Theorem for ∆ RQS.

a = RS ≈ √19.62 – 7.52 = 327.91 ≈ 18.108 √

Find the apothem a. The apothem is height RS of ∆PQR. Because ∆PQR is isosceles, altitude RS bisects QP .

Find the area of a regular polygon

Warm-Up ExercisesEXAMPLE 2

STEP 3 Find the area A of the table top.12A = aP Formula for area of regular polygon

≈ (18.108)(120)12 Substitute.

≈ 1086.5 Simplify.

Find the area of a regular polygon

So, the area you are covering with tiles is about 1086.5 square inches.

ANSWER

Warm-Up ExercisesEXAMPLE 3 Find the perimeter and area of a regular polygon

A regular nonagon is inscribed in a circle with radius 4 units. Find the perimeter and area of the nonagon.

SOLUTION360°

The measure of central JLK is , or 40°. Apothem LM bisects the central angle, so m KLM is 20°. To find the lengths of the legs, use trigonometric ratios for right ∆ KLM.

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Warm-Up ExercisesEXAMPLE 3

sin 20° = MKLK

sin 20° = MK4

4 sin 20° = MK

cos 20° = LMLK

cos 20° = LM4

4 cos 20° = LM

The regular nonagon has side length s = 2MK = 2(4 sin 20°) = 8 sin 20° and apothem a = LM = 4 cos 20°.

Find the perimeter and area of a regular polygon

Warm-Up ExercisesEXAMPLE 3 Find the perimeter and area of a regular polygon

So, the perimeter is P = 9s = 9(8 sin 20°) = 72 sin 20° ≈ 24.6 units,and the area is A = aP = (4 cos 20°)(72 sin 20°) ≈ 46.3 square units.

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ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

3.

Find the perimeter and the area of the regular polygon.

about 46.6 units, about 151.5 units2ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

4.

Find the perimeter and the area of the regular polygon.

70 units, about 377.0 units2ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

5.

30 3 52.0 units, about 129.9 units2ANSWER

Find the perimeter and the area of the regular polygon.

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

6. Which of Exercises 3–5 above can be solved using special right triangles?

Exercise 5ANSWER

Warm-Up ExercisesDaily Homework Quiz

1. Find the measure of the central angle of a regular polygon with 24 sides.

Find the area of each regular polygon.

ANSWER 15°

2.

ANSWER 110 cm2

Warm-Up ExercisesDaily Homework Quiz

3.

ANSWER 374.1 cm2

Warm-Up ExercisesDaily Homework Quiz

Find the perimeter and area of each regular polygon.

4.

ANSWER 99.4 in. ; 745.6 in.2

Warm-Up ExercisesDaily Homework Quiz

5.

ANSWER 22.6 m; 32 m2