Exercise 39 Page 509

Let's analyze the given quadrilateral to find the length of YW. Keep in mind, we have been told that the figure is a rectangle.

By the definition of a rectangle, we know that WXYZ has four right angles. Because of that, the triangle formed by WXZ is a right triangle. Recall the following theorem.

Pythagorean Theorem

For a right triangle with legs a and b and hypotenuse c, the following is true: a^2+b^2=c^2.

Using this theorem, we can write the following relation. XW^2 + WZ^2 = XZ^2 Let's substitute the given expressions for XW, WZ, and XZ into the equation. 3^2 + 4^2 = b^2 Let's solve it for b.

3^2+4^2 = b^2

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Solve for b

CalcPow

Calculate power

9+16=b^2

AddTerms

Add terms

25 = b^2

RearrangeEqn

Rearrange equation

b^2 = 25

RadicalEqn

sqrt(LHS)=sqrt(RHS)

b = ± 5

Since the length of a segment cannot be negative, we know that XZ=5. Recall that the diagonals of a rectangle are congruent. Therefore, their lengths are equal. YW=ZX ⇔ YW= 5