Answer:
1. B. x = 11
2. E. ![m\angle BEF=140^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20BEF%3D140%5E%7B%5Ccirc%7D)
3. D. 4.3 in
4. D. 40.8 ft
Step-by-step explanation:
1. By Angle Addition Postulate,
![m\angle KLM=m\angle KLV+m\angle VLM](https://tex.z-dn.net/?f=m%5Cangle%20KLM%3Dm%5Cangle%20KLV%2Bm%5Cangle%20VLM)
Since
![m\angle KLV=34^{\circ}\\ \\m\angle KLM=14x+19\\ \\m\angle VLM=12x+7,](https://tex.z-dn.net/?f=m%5Cangle%20KLV%3D34%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20KLM%3D14x%2B19%5C%5C%20%5C%5Cm%5Cangle%20VLM%3D12x%2B7%2C)
then
![14x+19=34+12x+7\\ \\14x-12x=34+7-19\\ \\2x=22\\ \\x=11](https://tex.z-dn.net/?f=14x%2B19%3D34%2B12x%2B7%5C%5C%20%5C%5C14x-12x%3D34%2B7-19%5C%5C%20%5C%5C2x%3D22%5C%5C%20%5C%5Cx%3D11)
2. By Angle Addition Postulate,
![m\angle FED=m\angle DEB+m\angle BEF](https://tex.z-dn.net/?f=m%5Cangle%20FED%3Dm%5Cangle%20DEB%2Bm%5Cangle%20BEF)
Since
![m\angle FED=14x+8\\ \\m\angle DEB=22^{\circ}\\ \\m\angle BEF=13x-3,](https://tex.z-dn.net/?f=m%5Cangle%20FED%3D14x%2B8%5C%5C%20%5C%5Cm%5Cangle%20DEB%3D22%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20BEF%3D13x-3%2C)
then
![14x+8=22+13x-3\\ \\14x-13x=22-3-8\\ \\x=11](https://tex.z-dn.net/?f=14x%2B8%3D22%2B13x-3%5C%5C%20%5C%5C14x-13x%3D22-3-8%5C%5C%20%5C%5Cx%3D11)
Therefore,
![m\angle BEF=(13\cdot 11-3)^{\circ}=(143-3)^{\circ}=140^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20BEF%3D%2813%5Ccdot%2011-3%29%5E%7B%5Ccirc%7D%3D%28143-3%29%5E%7B%5Ccirc%7D%3D140%5E%7B%5Ccirc%7D)
3. The area of trapezoid is
![A_{trapezoid}=\text{Midsegment}\times \text{Height}](https://tex.z-dn.net/?f=A_%7Btrapezoid%7D%3D%5Ctext%7BMidsegment%7D%5Ctimes%20%5Ctext%7BHeight%7D)
From the diagram,
![\text{Smaller base}=1.2\ in\\ \\\text{Bigger base}=4.2\ in,](https://tex.z-dn.net/?f=%5Ctext%7BSmaller%20base%7D%3D1.2%5C%20in%5C%5C%20%5C%5C%5Ctext%7BBigger%20base%7D%3D4.2%5C%20in%2C)
then
![\text{Midsegment}=\dfrac{1.2+4.2}{2}=2.7\ in](https://tex.z-dn.net/?f=%5Ctext%7BMidsegment%7D%3D%5Cdfrac%7B1.2%2B4.2%7D%7B2%7D%3D2.7%5C%20in)
Since the area of trapezoid is
then
![11.6\ in^2 =2.7\ in\times \text{Height}\\ \\\text{Height}=\dfrac{11.6\ in^2}{2.7\ in}\approx 4.3\ in](https://tex.z-dn.net/?f=11.6%5C%20in%5E2%20%3D2.7%5C%20in%5Ctimes%20%5Ctext%7BHeight%7D%5C%5C%20%5C%5C%5Ctext%7BHeight%7D%3D%5Cdfrac%7B11.6%5C%20in%5E2%7D%7B2.7%5C%20in%7D%5Capprox%204.3%5C%20in)
4. Use formula for the area of the circle to find the radius of the circle:
![A_{circle}=\pi r^2](https://tex.z-dn.net/?f=A_%7Bcircle%7D%3D%5Cpi%20r%5E2)
So,
![132.7=\pi r^2\\ \\r^2=\dfrac{132.7}{\pi}\\ \\r=\sqrt{\dfrac{132.7}{\pi}}\ ft](https://tex.z-dn.net/?f=132.7%3D%5Cpi%20r%5E2%5C%5C%20%5C%5Cr%5E2%3D%5Cdfrac%7B132.7%7D%7B%5Cpi%7D%5C%5C%20%5C%5Cr%3D%5Csqrt%7B%5Cdfrac%7B132.7%7D%7B%5Cpi%7D%7D%5C%20ft)
Now, find the circumference of the circle:
![C=2\pi r\\ \\C=2\pi \cdot \sqrt{\dfrac{132.7}{\pi}}=2\sqrt{132.7\pi}\approx 40.8\ ft](https://tex.z-dn.net/?f=C%3D2%5Cpi%20r%5C%5C%20%5C%5CC%3D2%5Cpi%20%5Ccdot%20%5Csqrt%7B%5Cdfrac%7B132.7%7D%7B%5Cpi%7D%7D%3D2%5Csqrt%7B132.7%5Cpi%7D%5Capprox%2040.8%5C%20ft)