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°
5
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
12
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.
9
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.
12
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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
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