Exercise 8 Page 504

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 ∠ Y, XY, and YZ.

Angle Measures

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

m ∠ Y + m ∠ X = 90 Since we know the measure of ∠ X, we can substitute it in our equation and find the measure of ∠ Y. m ∠ Y + 52 = 90 ⇔ m ∠ Y = 38

Side Lengths

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

The sine of ∠ Y is the ratio of the length of the leg opposite ∠ Y to the length of the hypotenuse. Sine=Opposite/Hypotenuse ⇒ sin 38^(∘) =8.5/x We can use the obtained equation to find the measure of XY. 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 38^(∘)=8.5/x

UseCalc

Use a calculator

0.615661 = 8.5/x

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Solve for x

MultEqn

LHS * x=RHS* x

0.615661x=8.5

DivEqn

.LHS /(0.615661).=.RHS /(0.615661).

x=13.806288

RoundDec

Round to 1 decimal place(s)

x ≈ 13.8

Therefore, the measure of XY is 13.8, rounded to nearest tenth. Finally, we can find the measure of YZ. To do it, we can use the Pythagorean Theorem. (XZ)^2 + (YZ)^2 = (XY)^2 Let's substitute the known lengths, XZ =8.5 and XY= 13.8, into this equation to find YZ.

(XZ)^2 + (YZ)^2 = (XY)^2

SubstituteII

XZ= 8.5, XY= 13.8

(8.5)^2 + (YZ)^2 = ( 13.8)^2

▼

Solve for YZ

CalcPow

Calculate power

72.25 + (YZ)^2=190.44

SubEqn

LHS-72.25=RHS-72.25

(YZ)^2= 118.19