Exercise 19 Page 542

We want to find the values of x, y, and z.

Notice that y is the hypotenuse of the given right triangle, and x and z are the legs. We will find their values one at a time.

Finding y

Since we know the lengths of a partial segment of the hypotenuse and the altitude, we will use the Geometric Mean (Altitude) Theorem to find the value of y.

We want to compare the theorem to the expressions in our figure. In our case, 17 is the length of the altitude, and 6 and ( y-6) are the lengths of the partial segments of the hypotenuse. AD/CD = CD/DB ⇔ 6/17 = 17/y-6 Now, we can find the value of y.

6/17 = 17/y-6

▼

Solve for y

MultEqn

LHS * 17=RHS* 17

6/17 * 17 = 17/y-6 * 17

FracMultDenomToNumber

a/17* 17 = a

6 = 17/y-6 * 17

MoveRightFacToNum

a/c* b = a* b/c

6 = 17 * 17/y-6

MultEqn

LHS * (y-6)=RHS* (y-6)

6(y-6) = 17 * 17/y-6 * (y-6)

FracMultDenomToNumber

a/(y-6)* (y-6) = a

6(y-6)=17 * 17

DivEqn

.LHS /6.=.RHS /6.

y-6=17 * 17/6

AddEqn

LHS+6=RHS+6

y=17 * 17/6+6

Multiply

Multiply

y=289/6+6

UseCalc

Use a calculator

y≈ 48.2+6

AddTerms

Add terms

y ≈ 54.2

Finding x and z

Let's go back to the given figure.

Since we know the lengths of the hypotenuse and its partial segment, we will use the Geometric Mean (Leg) Theorem to find the values of x and z.

We will start by finding the value of x, which corresponds to AC on this figure. AC = sqrt(AD * AB) ⇔ x=sqrt(6(54.2)) To find the value of x, we will evaluate the right-hand side.

x = sqrt(6 (54.2))

▼

Evaluate right-hand side

Multiply

Multiply

x = sqrt(325.2)

UseCalc

Use a calculator

x ≈ 18.0

Following the same reasoning, we can find z, which corresponds to CB. CB = sqrt(DB * AB) ⇔ z=sqrt(( 54.2- 6)54.2) Finally, we can evaluate the right-hand side to find the value of z.

z = sqrt((54.2-6)54.2)

▼

Evaluate right-hand side

SubTerm

Subtract term

z = sqrt(48.2* 54.2)

Multiply

Multiply

z = sqrt(2612.44)

UseCalc

Use a calculator

z ≈ 51.1