Exercise 39 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 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, 5+x is the length of the hypotenuse, AC is x+2, and AD is x. AB/AC=AC/AD ⇔ 5+x/x+2=x+2/x Now we can use the Cross Product Property to find the value of x.

5+x/x+2=x+2/x

CrossMult

Cross multiply

(5+x)x=(x+2)(x+2)

▼

Solve for x

Distr

Distribute x

5x+x^2=(x+2)(x+2)

ProdToPowTwoFac

a* a=a^2

5x+x^2=(x+2)^2

ExpandPosPerfectSquare

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

5x+x^2=x^2+4x+4

SubEqn

LHS-x^2=RHS-x^2

5x=4x+4

SubEqn

LHS-4x=RHS-4x

x=4