Length 26

The length of the median of a trapezoid is 10 inches. The median divides the trapezoid into two regions whose areas are in the ratio 3:5. What is the length of the shorter base?

Final Answer:

c =  5 in

Step-by-step explanation:

$$\begin{gathered} m=10\text{in} \\ {S}_1:S_2=3:5 \\ (a+c)/2=m \\ a+c=2\cdot m=20\text{in} \\ {h}_1=h_2 \\ {h}_1+h_2=h \\ {S}_1=h_1\cdot (a+m)/2 \\ {S}_2=h_2\cdot (c+m)/2=h_1\cdot (c+m)/2 \\ {h}_1\cdot (c+m)/2:h_1\cdot (a+m)/2=3:5 \\ (c+m):(a+m)=3:5 \\ (c+m)=\frac{3}{5}\cdot (a+m) \\ (a+c)/2=m \\ (c+10)=\frac{3}{5}\cdot (a+10) \\ (a+c)/2=10 \\ (c+10)=3/5\cdot (a+10) \\ (a+c)/2=10 \\ 3a-5c=20 \\ a+c=20 \\ Row2-\frac{1}{3}\cdot Row1\to Row2 \\ 2.67c=13.33 \\ c=\frac{13.33333333}{2.66666667}=5 \\ a=\frac{20+5c}{3}=\frac{20+5\cdot 5}{3}=15 \\ a=15 \\ c=5=5\text{in} \end{gathered}$$

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