Exercise 64 Page 740

We are given the following triangle and want to find the measures of x and y.

Since the given triangle is a right triangle, we will use the Pythagorean Theorem and Law of Sines. Let's find the measures of x and y one at a time.

Finding x

Let's begin by color coding the opposite angles and sides in the given triangle.

Since △ ABC is a right triangle, we can use the formula of the Pythagorean theorem. c^2=a^2+b^2 Looking at the triangle, the length of the hypotenuse is c=6, a=3, and b= x. Let's substitute these values into the formula to find the value of x.

c^2=a^2+b^2

SubstituteValues

Substitute values

6^2=3^2+ x^2

▼

Solve for x

CalcPow

Calculate power

36=9+x^2

SubEqn

LHS-9=RHS-9

27=x^2

RearrangeEqn

Rearrange equation

x^2=27

SqrtEqn

sqrt(LHS)=sqrt(RHS)

x=sqrt(27)

Therefore, the value of x is sqrt(27). Now, we will find the measure of ∠ A since this will help us in the next calculations. To find the measure of ∠ A, we will use the Law of Sines. sin (A)/a=sin (C)/c Let's substitute a=3, c= 6, and m ∠ C = 90 to isolate sin A.

sin (A)/a=sin (C)/c

SubstituteValues

Substitute values

sin (A)/3=sin (90)/6

▼

Solve for sin A

MultEqn

LHS * 3=RHS* 3

sin (A)=3 sin (90)/6

ReduceFrac

a/b=.a /3./.b /3.

sin (A)=sin (90)/2

Now we can use the inverse sine ratio to find m ∠ A.

m ∠ A = sin ^(-1)(sin (90)/2)

UseCalc

Use a calculator

m ∠ A = 30

Therefore, m∠ A is 30^(∘).

Finding y

To find the value of y, let's begin by color coding the opposite angles and sides in the given triangle. It will help us use the Law of Sines.

First, we must find the measure of ∠ D. Since the △ ABD is a right triangle, we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to 180. m ∠ D+ m∠ A+ m∠ B&= 180 &⇕ m ∠ D+ 30+ 90&= 180 m ∠ D+ 120&= 180 m ∠ D&= 180-120 m ∠ D&= 60 Now that we know the measure of ∠ D, we can find y by using the Law of Sines. sin ( C)/y =sin (D)/d Let's substitute d=3, m ∠ C = 90, and m ∠ D = 60 to isolate y.

sin (C)/y =sin (D)/d

SubstituteValues

Substitute values

sin ( 90)/y =sin ( 60)/3

▼

Solve for y

CrossMult

Cross multiply

3sin (90) =ysin (60)

RearrangeEqn

Rearrange equation

ysin(60)=3sin(90)

DivEqn

.LHS /sin(60).=.RHS /sin(60).

y=3sin(90)/sin(60)

UseCalc

Use a calculator

y=3.464101...

RoundDec

Round to 1 decimal place(s)

y ≈ 3.5

Therefore, the value of y rounded to the nearest tenth is 3.5.

Completing the Triangle

With all of the side measures, we can complete our diagram.