Exercise 23 Page 542

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

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

Finding z and x

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

We will start by finding the value of z, which corresponds to AD on this figure. Notice that since AD and DB are partial segments of the hypotenuse, the sum of their lengths is equal to AB. AB/CB = CB/DB ⇔ z+4/12 = 12/4 Now, we can find the value of z.

z+4/12 = 12/4

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Solve for z

CalcQuot

Calculate quotient

z+4/12 = 3

MultEqn

LHS * 12=RHS* 12

z+4/12 * 12 = 3 * 12

FracMultDenomToNumber

a/12* 12 = a

z + 4 = 3 * 12

Multiply

Multiply

z+4 = 36

SubEqn

LHS-4=RHS-4

z = 32

Having found the value of z, we can find x, which corresponds to AC. AC = sqrt(AD * AB) ⇔ x = sqrt(32(32+4)) We will evaluate the right-hand side to find the value of x.

x = sqrt(32 (32+4))

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Evaluate right-hand side

AddTerms

Add terms

x = sqrt(32(36))

SplitIntoFactors

Split into factors

x = sqrt(32 * 2 * 9 * 2)

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

x = sqrt(32 * 2) * sqrt(9) * sqrt(2)

Multiply

Multiply

x = sqrt(64) * sqrt(9) * sqrt(2)

CalcRoot

Calculate root

x = 8 * 3 * sqrt(2)

Multiply

Multiply

x = 24sqrt(2)

Using a calculator, we can express x as about 33.9.

Finding y

Let's go back to the given figure.

Since we know the lengths of the partial segments of the hypotenuse divided by the altitude we will use the Geometric Mean (Altitude) Theorem to find the value of y.

Let's compare the theorem to the expressions in our figure. In our case, y is the length of the altitude, and 32 and 4 are the lengths of the partial segments of the hypotenuse. CD = sqrt(AD * DB) ⇔ y=sqrt(32* 4) Finally, we can evaluate the right-hand side to find the value of y.

y = sqrt(32 * 4)

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Evaluate right-hand side

SplitIntoFactors

Split into factors

y = sqrt(32 * 2 * 2)

ProdSqrt

sqrt(a)*sqrt(b)=sqrt(a* b)

y = sqrt(32 * 2) * sqrt(2)

Multiply

Multiply

y = sqrt(64) * sqrt(2)

CalcRoot

Calculate root

y = 8sqrt(2)

Using a calculator, we can express y as about 11.3.