Exercise 5 Page 623

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

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

Finding x

To find the value of x, we will use the Geometric Mean (Altitude) Theorem since we know the lengths of both partial segments of the hypotenuse divided by the altitude.

We want to compare the theorem to the expressions in our figure. In our case, x is the length of the altitude, and 3 and 12 are the lengths of the partial segments of the hypotenuse. CD = sqrt(AD * DB) ⇔ x=sqrt(3* 12) Now, we can evaluate the right-hand side to find the value of x.

x = sqrt(3 * 12)

▼

Evaluate right-hand side

Multiply

Multiply

x = sqrt(36)

CalcRoot

Calculate root

x = 6

Finding y and z

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 (Leg) Theorem to find the values of y and z.

We will start by finding the value of y, which corresponds to AC on this figure. AC=sqrt(AD * AB) ⇔ y=sqrt(3(12+3)) Now we can evaluate the right-hand side to find y.

y = sqrt(3 (12+3))

▼

Evaluate right-hand side

AddTerms

Add terms

y = sqrt(3(15))

SplitIntoFactors

Split into factors

y = sqrt(3 * 3 * 5)

SqrtProd

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

y = sqrt(3 * 3) * sqrt(5)

Multiply

Multiply

y = sqrt(9) * sqrt(5)

CalcRoot

Calculate root

y = 3sqrt(5)

Using a calculator, we can express y as about 6.7. Following the same reasoning, we can find z, which corresponds to CB. CB = sqrt(DB * AB) ⇔ z=sqrt(12(12+3)) Finally, we can evaluate the right-hand side to find the value of z.

z = sqrt(12 (12+3))

▼

Evaluate right-hand side

AddTerms

Add terms

z = sqrt(12(15))

SplitIntoFactors

Split into factors

z = sqrt(4 * 3 * 3 * 5)

SqrtProd

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

z = sqrt(4) * sqrt(3 * 3) * sqrt(5)

CalcRoot

Calculate root

z = 2 * 3 sqrt(5)

Multiply

Multiply

z = 6sqrt(5)

Using a calculator, we can rewrite z as about 13.4.