Exercise 18 Page 505

Let's analyze the given right triangle.

We will find the missing measures one at a time. In this case, this means that we want to find m ∠ R, RT, and RS.

Angle Measures

To find m∠ T, recall that the acute angles of a right triangle are complementary. Therefore, m ∠ T and m ∠ R add to 90.

m ∠ T + m ∠ R = 90 Since we know the measure of ∠ R, we can substitute it in our equation and find the measure of ∠ T. m ∠ T + 57 = 90 ⇔ m ∠ T = 33

Side Lengths

We can find the measure of RT using a sine ratio.

The sine of ∠ R is the ratio of the length of the leg opposite ∠ R to the length of the hypotenuse. Sine=Opposite/Hypotenuse ⇒ sin 57^(∘) =15/x We can use the obtained equation to find the measure of RT. We have to use the calculator to find the value of the given sine. Then, we will substitute it into the equation and solve for x.

sin 57^(∘)=15/x

UseCalc

Use a calculator

0.838671 = 15/x

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Solve for x

MultEqn

LHS * x=RHS* x

0.838671x=15

DivEqn

.LHS /(0.838671).=.RHS /(0.838671).

x=17.885440

RoundDec

Round to 1 decimal place(s)

x ≈ 17.9

Therefore, the measure of RT is 17.9, rounded to nearest tenth. Finally, we can find the measure of RS. To do it, we can use the Pythagorean Theorem. (RS)^2 + (ST)^2 = (RT)^2 Let's substitute the known lengths, ST = 15 and RT= 17.9, into this equation to find YZ.

(RS)^2 + (ST)^2 = (RT)^2

SubstituteII

ST= 15, RT= 17.9

(RS)^2+ 15^2 = ( 17.9)^2

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Solve for RS

CalcPow

Calculate power

(RS)^2+ 225=320.41

SubEqn

LHS-225=RHS-225

(RS)^2= 95.41