Exercise 96 Page 583

Practice makes perfect

a Let's draw ⊙ B. Note that if mAC=80^(∘), the central angle that makes up this arc will have the same measure.

Examining the diagram, we notice that the chord is a side in an isosceles triangle with a vertex angle of 80^(∘). Using the Law of Cosines, we can find the measure of AC.

a^2=b^2+c^2-2bc cos B

SubstituteValues

Substitute values

(AC)^2=10^2+10^2-2(10)(10) cos 80^(∘)

▼

Simplify right-hand side

CalcPow

Calculate power

(AC)^2=100+100-2(10)(10) cos 80^(∘)

SqrtEqn

sqrt(LHS)=sqrt(RHS)

AC=± sqrt(100+100-2(10)(10) cos 80^(∘))

b > 0

AC=sqrt(100+100-2(10)(10) cos 80^(∘))

UseCalc

Use a calculator

AC=12.85575...

RoundDec

Round to 2 decimal place(s)

AC≈ 12.85

The length of chord AC is about 12.85 units.

b Let's illustrate the circle.

From the given information we can determine the length of KF as 15-6=9 units. Let's also label GK as x.

The lengths of the resulting segments forms sides in two similar triangles. With this information, we can write the following equation. x/9=6/3 Let's solve this equation for x.

x/9=6/3

CalcQuot

Calculate quotient

x/9=2

MultEqn

LHS * 9=RHS* 9

x=18

The length of GK is 18 units.

Hints

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