Exercise 20 Page 646

Practice makes perfect

a We are given a regular pentagon with center O and radius 10 inches. We see that ∠ POQ is the central angle of the pentagon.

The formula to find the measure of a central angle of a regular n-gon is 360n. A pentagon has 5 sides, so let's substitute 5 for n and solve for the measure of the central angle. 360/5=72 The measure of a central angle of a pentagon is 72^(∘), so m∠ POQ=72^(∘).

b We see that OX is the altitude to the base of the isosceles triangle POQ.

By definition, OX is also the apothem of the pentagon and bisects ∠ POQ. Let's use this to find the measure of ∠ POX.

m∠ POX=1/2m∠ POQ

▼

Solve for m∠ POX

Substitute

m∠ POQ= 72

m∠ POX = 1/2( 72)

MoveRightFacToNum

a/c* b = a* b/c

m∠ POX = 72/2

CalcQuot

Calculate quotient

m∠ POX = 36

Therefore, m∠ POX=36^(∘).

c Let a represent the length of the apothem.

To do find a, we will use the cosine ratio of ∠ POX. cos 36^(∘) = a/10 We will now solve for a.

cos 36^(∘) = a/10

▼

Solve for a

MultEqn

LHS * 10=RHS* 10

10 * cos 36 ^(∘) =a

RearrangeEqn

Rearrange equation

a=10 * cos 36 ^(∘)

UseCalc

Use a calculator

a=8.090169 ...

RoundDec

Round to 1 decimal place(s)

a≈ 8.1

Therefore, OX is about 8.1 inches.

d Since the apothem bisects PQ, we will first find PX. Let x be the length of PX.

To do find x, we will use the sine ratio of ∠ POX. sin 36^(∘) = x/10 We will now solve for x.

sin 36^(∘) = x/10

▼

Solve for x

MultEqn

LHS * 10=RHS* 10

10 * sin 36 ^(∘) =x

RearrangeEqn

Rearrange equation

x=10 * sin 36 ^(∘)

UseCalc

Use a calculator

x=5.877852 ...

RoundDec

Round to 1 decimal place(s)

x≈ 5.9

We know that PQ is twice the length of PX. We found that PX is about 5.9 inches, so PQ is twice this length.

PQ = 2(PX)

▼

Solve for PQ

Substitute

PX= 5.9

PQ = 2( 5.9)

Multiply

PQ=11.8

Therefore, PQ=11.8 inches.

e From Part D, we know that the regular pentagon PQRST has a side length 11.8 inches. To calculate the perimeter p of the pentagon, we will multiply the number of sides 5 by the side length.

p= 5 * 11.8 =59 The perimeter of PQRST is about 59 inches.

f We have shown that the apothem has a length of 8.1 inches and the perimeter of the pentagon is 59 inches. We can find the area of PQRST by substituting a= 8.1 and p=59 into the formula for the area of a regular polygon, A= 12ap.

A=1/2ap

SubstituteII

a= 8.1, p= 59

A=1/2* 8.1* 59

Multiply

A=238.95

RoundDec

Round to 1 decimal place(s)

A=239.0

The area of the PQRST is about 239.0 square inches.