Math worksheet for calculating the midsegment of trapezoids, including problems with given side lengths and algebraic expressions.

Here are the step-by-step solutions for the worksheet.

Part 1: Find the midsegment of the given trapezoid

The rule for finding the midsegment is simple: Add the lengths of the top and bottom bases together, then divide by 2.
Formula: $\text{Midsegment} = \frac{\text{Base}_1 + \text{Base}_2}{2}$

a)
* Top base ($XY$) = $12\text{ m}$
* Bottom base ($ZW$) = $18\text{ m}$
* Calculation: $(12 + 18) / 2 = 30 / 2 = 15$
* MN = 15 m

b)
* Top base ($QR$) = $6.5\text{ cm}$
* Bottom base ($UT$) = $12.5\text{ cm}$
* Calculation: $(6.5 + 12.5) / 2 = 19 / 2 = 9.5$
* VS = 9.5 cm

c)
* Top base ($AB$) = $34\text{ yd}$
* Bottom base ($ED$) = $48\text{ yd}$
* Calculation: $(34 + 48) / 2 = 82 / 2 = 41$
* FC = 41 yd

d)
* This trapezoid is turned on its side. The parallel sides (bases) are vertical.
* Left base ($WZ$) = $8\text{ yd}$
* Right base ($RQ$) = $26\text{ yd}$
* Calculation: $(8 + 26) / 2 = 34 / 2 = 17$
* TS = 17 yd

e)
* This trapezoid is also turned. The parallel sides are vertical.
* Left base ($EF$) = $18.8\text{ ft}$
* Right base ($AB$) = $16.6\text{ ft}$
* Calculation: $(18.8 + 16.6) / 2 = 35.4 / 2 = 17.7$
* DC = 17.7 ft

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Part 2: Solve for 'x' using the midsegment theorem

The rule here is: The midsegment length is equal to the average of the two bases.
Equation: $\text{Midsegment} = \frac{\text{Top} + \text{Bottom}}{2}$
*(Or you can multiply the midsegment by 2 to equal the sum of the bases)*

a)
* Top = $10$, Bottom = $24$, Midsegment = $2x + 10$
* Equation: $2x + 10 = \frac{10 + 24}{2}$
* Simplify the right side: $\frac{34}{2} = 17$
* New Equation: $2x + 10 = 17$
* Subtract 10 from both sides: $2x = 7$
* Divide by 2: $x = 3.5$
* x = 3.5

b)
* Top = $61$, Bottom = $11$, Midsegment = $5x + 11$
* Equation: $5x + 11 = \frac{61 + 11}{2}$
* Simplify the right side: $\frac{72}{2} = 36$
* New Equation: $5x + 11 = 36$
* Subtract 11 from both sides: $5x = 25$
* Divide by 5: $x = 5$
* x = 5

c)
* Top = $22$, Bottom = $48$, Midsegment = $x + 12$
* Equation: $x + 12 = \frac{22 + 48}{2}$
* Simplify the right side: $\frac{70}{2} = 35$
* New Equation: $x + 12 = 35$
* Subtract 12 from both sides: $x = 23$
* x = 23

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Final Answer:
1)
a) MN = 15 m
b) VS = 9.5 cm
c) FC = 41 yd
d) TS = 17 yd
e) DC = 17.7 ft

2)
a) x = 3.5
b) x = 5
c) x = 23