Centimeters - trapezoid

The area of the trapezoid is 132 cm². The difference in the length of both bases is 6 cm, and the height is 2 cm longer than the shorter base. Find the height of the trapezoid in centimeters.

Correct answer:

h =  11 cm

Step-by-step explanation:

$$\begin{gathered} S=132{\text{cm}}^2 \\ a-c=6\text{cm} \\ h=2+c \\ S=\frac{a+c}{2}\cdot h \\ S=\frac{(6+c)+c}{2}\cdot (2+c) \\ 2\cdot S=((6+c)+c)\cdot (2+c) \\ 2\cdot 132=((6+c)+c)\cdot (2+c) \\ -2c^2-10c+252=0 \\ 2c^2+10c-252=0 \\ 2...\text{ prime number} \\ 10=2\cdot 5 \\ 252=2^2\cdot 3^2\cdot 7 \\ \text{GCD}(2,10,252)=2 \\ {c}^2+5c-126=0 \\ p=1;q=5;r=-126 \\ D=q^2-4pr=5^2-4\cdot 1\cdot (-126)=529 \\ D>0 \\ {c}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{-5\pm \sqrt{529}}{2} \\ {c}_{1,2}=\frac{-5\pm 23}{2} \\ {c}_{1,2}=-2.5\pm 11.5 \\ {c}_1=9 \\ {c}_2=-14 \\ c=c_1=9\text{cm} \\ a=6+c=6+9=15\text{cm} \\ h=2+c=2+9=11\text{cm} \\ \text{ Verifying Solution: } \\ {S}_2=\frac{a+c}{2}\cdot h=\frac{15+9}{2}\cdot 11=132{\text{cm}}^2 \end{gathered}$$

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