Exercise 29 Page 542

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

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

Finding $x$ and $z$

Since we know the lengths of one leg and of the hypotenuse, 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

D B

on this figure. Notice that the difference between

A B

and

D B

is

A D .

\frac{A B}{A C} = \frac{A C}{A D} \Leftrightarrow \frac{2 3}{3} = \frac{3}{2 3 - x}

Now, we can find the value of

x .

\frac{2 3}{3} = \frac{3}{2 3 - x}

▼

Solve for

x

MultEqn

$LHS \cdot 3 = RHS \cdot 3$

\frac{2 3}{3} \cdot 3 = \frac{3}{2 3 - x} \cdot 3

FracMultDenomToNumber

$\frac{a}{3} \cdot 3 = a$

23 = \frac{3}{2 3 - x} \cdot 3

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

23 = \frac{3 \cdot 3}{2 3 - x}

MultSqrt

$a \cdot a = a$

23 = \frac{3}{2 3 - x}

MultEqn

$LHS \cdot (23 - x) = RHS \cdot (23 - x)$

23 \cdot (23 - x) = \frac{3}{2 3 - x} \cdot (23 - x)

FracMultDenomToNumber

$\frac{a}{( 2 3 - x )} \cdot (23 - x) = a$

23 (23 - x) = 3

Distr

Distribute $23$

12 - 23 x = 3

SubEqn

$LHS - 12 = RHS - 12$

- 23 x = - 9

MultEqn

$LHS \cdot (- 1) = RHS \cdot (- 1)$

23 x = 9

DivEqn

$LHS / 23 = RHS / 23$

x = \frac{9}{2 3}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

x = \frac{9 \cdot 3}{2 \cdot 3 \cdot 3}

MultSqrt

$a \cdot a = a$

x = \frac{9 \cdot 3}{2 \cdot 3}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

x = \frac{3 3}{2}

Using a calculator, we can express

x,

which corresponds to

D B,

as about

2.6 .

Having found the value of

x,

we can find

z,

which corresponds to

C B .

C B = D B \cdot A B ⇕ z = \frac{3 3}{2} \cdot 23

We will evaluate the right-hand side to find the value of

z .

z = \frac{3 3}{2} \cdot 23

▼

Evaluate right-hand side

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

z = \frac{3 3 \cdot 2 3}{2}

Multiply

z = \frac{6 3 \cdot 3}{2}

MultSqrt

$a \cdot a = a$

z = \frac{6 \cdot 3}{2}

Multiply

z = \frac{1 8}{2}

CalcQuot

Calculate quotient

z = 9

CalcRoot

Calculate root

z = 3

Finding $y$

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 (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

(23 - \frac{3 3}{2})

and

\frac{3 3}{2}

are the lengths of the partial segments of the hypotenuse.

C D = A D \cdot D B ⇕ y = (23 - \frac{3 3}{2}) \frac{3 3}{2}

Finally, we can evaluate the right-hand side to find

y .

y = (23 - \frac{3 3}{2}) \frac{3 3}{2}

▼

Evaluate right-hand side

NumberToFrac

$a = \frac{2 \cdot a}{2}$

y = (\frac{4 3}{2} - \frac{3 3}{2}) \frac{3 3}{2}

SubFrac

Subtract fractions

y = (\frac{4 3 - 3 3}{2}) \frac{3 3}{2}

SubTerm

Subtract term

y = (\frac{3}{2}) \frac{3 3}{2}

Multiply

y = \frac{9}{4}

SqrtQuot

$\frac{a}{b} = \frac{a}{b}$

y = \frac{9}{4}

CalcRoot

Calculate root

y = \frac{3}{2}