Question

example 2
trapezoid rhwa with midsegment \\(\overline{uk}\\) is shown. find the values of
x and y if \\( ru = -2x - 2y \\), \\( au = 10 \\), \\( rh = 4x \\), \\( uk = 14 \\), and
\\( aw = x + 13 \\)?

Explanation:

Step1: Use trapezoid midsegment formula

The midsegment of a trapezoid equals half the sum of the two bases. So:
$$UK = \frac{RH + AW}{2}$$
Substitute the given values:
$$14 = \frac{4x + x + 13}{2}$$

Step2: Solve for x

Multiply both sides by 2:
$$28 = 5x + 13$$
Subtract 13 from both sides:
$$15 = 5x$$
Divide by 5:
$$x = 3$$

Step3: Use midpoint property for RU=AU

U is the midpoint of RA, so $RU = AU$. Substitute values:
$$-2x - 2y = 10$$

Step4: Substitute x=3, solve for y

Plug $x=3$ into the equation:
$$-2(3) - 2y = 10$$
$$-6 - 2y = 10$$
Add 6 to both sides:
$$-2y = 16$$
Divide by -2:
$$y = -8$$

Answer:

$x=3$, $y=-8$