Exercise 40 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 the length of the leg adjacent to the second segment. 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, 18+x is the length of the hypotenuse, x is the length of the shorter segment of the hypotenuse, and 12 is the length of the shorter leg. AB/AC=AC/AD ⇔ 18+x/12=12/x Now we can use the Cross Product Property to find the value of x.

18+x/12=12/x

CrossMult

Cross multiply

(18+x)x=12* 12

▼

Solve for x

Distr

Distribute x

18x+x^2=12* 12

Multiply

Multiply

18x+x^2=144

SubEqn

LHS-144=RHS-144

18x+x^2-144=0

CommutativePropAdd

Commutative Property of Addition

x^2+18x-144=0

WriteDiff

Write as a difference

x^2+24x-6x-144=0

FactorOut

Factor out x & - 6

x(x+24)-6(x+24)=0

FactorOut

Factor out (x+24)

(x+24)(x-6)=0

ZeroProdProp

Use the Zero Product Property

lcx+24=0 & (I) x-6=0 & (II)

SubEqn

(I):LHS-24=RHS-24

lx=- 24 x-6=0

AddEqn

(II):LHS+6=RHS+6

lx=- 24 x=6

We have that x is either equal to - 24 or 6. Note that since x represents a side length, it must be positive. Therefore, the only possible value of x is 6.