Question

1. trapezoid whnt is shown where r is the midpoint of \\( \overline{hn} \\), y is the midpoint of \\( \overline{wt} \\). \\( wh = 11x - 2.5 \\), \\( nh = 9x + 1.5 \\), \\( tn = 3x + 17.5 \\), \\( ry = 9x - 4.5 \\) and \\( wt = 6x + 8.5 \\)

a. what is \\( x \\)? \\( x = \square \\)
b. what is the measurement of \\( yt \\)?
\\( \circ 22.25 \\)
\\( \circ 11.8 \\)
\\( \circ 23.6 \\)
\\( \circ 37.3 \\)
\\( \circ 36 \\)
\\( \circ 44.5 \\)

1. the regulations of a paper airplane contest state that all paper airplanes must be in the shape of an isosceles trapezoid. jaxson used computer software to sketch a trapezoid, as shown.

what is the measure of \\( \angle ade \\)?
\\( \circ 42^\circ \\)
\\( \circ 51^\circ \\)
\\( \circ 40^\circ \\)
\\( \circ 54^\circ \\)

Explanation:

Problem 1

Step1: Set midsegment equation

In a trapezoid, the midsegment length is the average of the two bases. So:
$$RY = \frac{WH + TN}{2}$$
Substitute the given expressions:
$$9x - 4.5 = \frac{(11x - 2.5) + (3x + 17.5)}{2}$$

Step2: Simplify right-hand side

Combine like terms in the numerator:
$$9x - 4.5 = \frac{14x + 15}{2}$$
Multiply both sides by 2:
$$18x - 9 = 14x + 15$$

Step3: Solve for x

Subtract $14x$ from both sides:
$$4x - 9 = 15$$
Add 9 to both sides:
$$4x = 24$$
Divide by 4:
$$x = 6$$

Step4: Calculate length of YT

Y is the midpoint of $\overline{WT}$, so $YT = \frac{WT}{2}$. First find $WT$:
$$WT = 6x + 8.5 = 6(6) + 8.5 = 36 + 8.5 = 44.5$$
Then:
$$YT = \frac{44.5}{2} = 22.25$$

Step1: Set equal leg lengths

It is an isosceles trapezoid, so the non-parallel legs are equal:
$$7x + 1 = 5x + 11$$

Step2: Solve for x

Subtract $5x$ from both sides:
$$2x + 1 = 11$$
Subtract 1 from both sides:
$$2x = 10$$
Divide by 2:
$$x = 5$$

Step3: Calculate $\angle ADE$

Substitute $x=5$ into the angle expression:
$$\angle ADE = 9x - 3 = 9(5) - 3 = 45 - 3 = 42^\circ$$

Answer:

A. $x = 6$
B. 22.25

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Problem 2