Exercise 20 Page 542

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

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

Finding y

Since we know the lengths of both partial segments of the hypotenuse divided by 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, y is the length of the altitude, and 4 and 6 are the lengths of the partial segments of the hypotenuse. CD = sqrt(AD * DB) ⇔ y=sqrt(4 * 6) Now, we can find the value of y.

y = sqrt(4 * 6)

▼

Evaluate right-hand side

ProdSqrt

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

y = sqrt(4) * sqrt(6)

CalcRoot

Calculate root

y = 2sqrt(6)

Using a calculator, we can express y as about 4.9.

Finding x and z

Let's go back to the given figure.

Since we know the lengths of both segments of the hypotenuse divided by the altitude, 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(4(4+6)) Now we can evaluate the right-hand side to find x.

x = sqrt(4 (4+6))

▼

Evaluate right-hand side

AddTerms

Add terms

x = sqrt(4(10))

SqrtProd

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

x = sqrt(4) * sqrt(10)

CalcRoot

Calculate root

x = 2 sqrt(10)

Using a calculator, we can express x as about 6.3. Following the same reasoning, we can find z, which corresponds to CB. CB = sqrt(DB * AB) ⇔ z=sqrt(6(4+6)) Finally, we can evaluate the right-hand side to find the value of z.

z = sqrt(6 (4+6))

▼

Evaluate right-hand side

AddTerms

Add terms

z = sqrt(6(10))

SplitIntoFactors

Split into factors

z = sqrt(2 * 3 * 2 * 5)

CommutativePropMult

Commutative Property of Multiplication

z = sqrt(2 * 2 * 3 * 5)

SqrtProd

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

z = sqrt(2 * 2) * sqrt(3 * 5)

Multiply

Multiply

z = sqrt(4) * sqrt(15)

CalcRoot

Calculate root

z = 2sqrt(15)

Using a calculator, we can rewrite z as about 7.7.