Exercise 21 Page 465

Let's analyze the given right triangle so that we can find the values of x and y.

Consider the bigger triangle. We know the hypotenuse, an expression for the altitude, and an expression for one of the legs. Therefore, we can use the corollaries of the Right Triangle Similarity Theorem to write two proportions. Let's do it!

Finding x

To find the value of x, we will use the following corollary of the Right Triangle Similarity Theorem.

Let's compare the theorem's corollary to the expressions in our figure. In our case, AD is 7, DB is 9, and x is the altitude of the bigger triangle CD. AD/CD=CD/DB ⇔ 7/x=x/9 Now we can use the Cross Product Property to find the value of x.

7/x=x/9

CrossMult

Cross multiply

7* 9 = x* x

▼

Solve for x

RearrangeEqn

Rearrange equation

x* x=7* 9

ProdToPowTwoFac

a* a=a^2

x^2=7* 9

SqrtEqn

sqrt(LHS)=sqrt(RHS)

x=sqrt(7* 9)

SqrtProd

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

x=sqrt(7)*sqrt(9)

CalcRoot

Calculate root

x=3sqrt(7)

Note that, when solving the equation, we only considered the principal root. The reason for this is that, since x represents a side length, it must be positive. Therefore, the value of x is 3sqrt(7).

Finding y

To find the value of y, we will use the following corollary.

In a similar manner, let's compare the second theorem's corollary to the expressions in our figure. Consider the bigger triangle. The length of the longer leg is y, the hypotenuse is 9+7= 16, and the longer segment of the hypotenuse is 9. AB/CB=CB/DB ⇔ 16/y=y/9 Now we can use the Cross Product Property to find the value of y.

16/y=y/9

CrossMult

Cross multiply

16* 9 = y* y

▼

Solve for y

Multiply

Multiply

144 = y* y

ProdToPowTwoFac

a* a=a^2

144=y^2

RearrangeEqn

Rearrange equation

y^2=144

SqrtEqn

sqrt(LHS)=sqrt(RHS)

y=sqrt(144)

CalcRoot

Calculate root

y=12

Again, when solving the equation, we only considered the principal root. The reason for this is that, since y also represents a side length, it must be positive. The value of y is 12.