Exercise 41 Page 466

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

Consider the bigger triangle. We know the length of one of the segments of the hypotenuse, an expression for the length of the second segment of the hypotenuse, and an expression for the length of the altitude. Therefore, we can use a corollary of the Right Triangle Similarity Theorem to write a proportion.

Let's compare the theorem's corollary to the expressions in our figure. In our case, x is the length of the shorter segment of the hypotenuse, 20 is the length of the longer segment of the hypotenuse, and x+5 is the length of the altitude. AD/CD=CD/DB ⇔ x/x+5=x+5/20 Now we can use the Cross Product Property to find the value of x.

x/x+5=x+5/20

CrossMult

Cross multiply

20x = (x+5)(x+5)

▼

Solve for x

ProdToPowTwoFac

a* a=a^2

20x=(x+5)^2

ExpandPosPerfectSquare

(a+b)^2=a^2+2ab+b^2

20x=x^2+10x+25

SubEqn

LHS-20x=RHS-20x

0=x^2-10x+25

RearrangeEqn

Rearrange equation

x^2-10x+25=0

FacNegPerfectSquare

a^2-2ab+b^2=(a-b)^2

(x-5)^2=0

PowToProdTwoFac

a^2=a* a

(x-5)(x-5)=0

ZeroProdProp

Use the Zero Product Property

lcx-5=0 & (I) x-5=0 & (II)

(I), (II): LHS+5=RHS+5

lx=5 x=5

Because both possible answers are x=5, we know that x must be equal to 5.