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# geometry

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Two 3 * 4 rectangles overlap. Find the area of the overlapping region (which is shaded).

(Link for picture: https://ibb.co/0q5hhpm)

Pic added by Melody:

Oct 18, 2022
edited by Melody  Oct 18, 2022

### 3+0 Answers

#2
+118132
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Two 3 * 4 rectangles overlap. Find the area of the overlapping region (which is shaded green).

$$tan\alpha=\frac{3}{4}=0.75\\ \alpha=atan0.75\approx 36.87^0\\ 2\alpha\approx 73.74^0\\ sin73.74=\frac{3}{y}\\ y\approx \frac{3}{sin73.74}\approx3.125\\ Green Area=2*0.5*y*y*sin(180-2\alpha)\\ Green Area=y*y*sin(2\alpha)\\ Green Area=y*y*\frac{3}{y}\\ Green Area=y*3\\ Green Area = 3*3.125\\ Green Area \approx 9.375u^2\\$$

LaTex:

tan\alpha=\frac{3}{4}=0.75\\
\alpha=atan0.75\approx 36.87^0\\
2\alpha\approx 73.74^0\\
sin73.74=\frac{3}{y}\\
y\approx \frac{3}{sin73.74}\approx3.125\\
Green Area=2*0.5*y*y*sin(180-2\alpha)\\
Green Area=y*y*sin(2\alpha)\\
Green Area=y*y*\frac{3}{y}\\
Green Area=y*3\\
Green Area = 3*3.125\\
Green Area \approx  9.375u^2\\

Oct 18, 2022
#3
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Melody, I need the exact answer, not just the approximate answer.

Guest Oct 19, 2022
#4
+118132
+1

You want it exact, you could have done it for yourself.

But here it is.

$$tan\alpha=\frac{3}{4}=0.75\\ \alpha=atan0.75\\ sin(2\alpha )=\frac{3}{y}\\ y= \frac{3}{sin(2\alpha)}\\ Green Area=2*0.5*y*y*sin(180-2\alpha)\\ Green Area=y*y*sin(2\alpha)\\ Green Area=y*y*\frac{3}{y}\\ Green Area=3y \\Green Area=\frac{9}{sin(2\alpha)} \\Green Area=\frac{9}{2sin\alpha cos\alpha} \\Green Area=\frac{9}{2*\frac{3}{5}*\frac{4}{5}}\\ \\Green Area=\frac{9*25}{24}\\ \\Green Area=9.375$$

Melody  Oct 19, 2022