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# Rectangles

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Two rectangles overlap, as shown below. Find the area of the overlapping region (which is shaded) if AB = BE = 2 and AD = ED = 4. Nov 30, 2022

#1
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Hello Guest!

The area of a rectangle of length l and width w is given by the multiplication of the dimensions, as follows:

A = lw.

The dimensions of the right triangle as follows:

Leg x.

Leg 2.

Hence the remaining leg on the overlapping region is:

4 - x, as AD = 4.

By symmetry, the other dimension of the overlapping region is also of:

4 - x.

Being also the hypotenuse of the right triangles.

The value of x can be found applying the Pythagorean Theorem as follows:

x² + 2² = (4 - x)²

x² + 4 = 16 - 8x + x²

8x = 12

x = 1.5.

Then the two dimensions of the shaded region are:

4 - 1.5 = 2.5.

Meaning that the area is of:

A = 2.5 x 2.5 = 6.25 units squared.

Nov 30, 2022
#2
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Imcool:

The shaded area is NOT a rectangle or square! The bottom triangle with base CD ==2 and its height is ==1.5. Its area ==[2 x 1.5] / 2 ==1.5 u^2

Do the same with the similar triangle at the top with the base AB ==2

Add the area of the 2 triangles ==1.5 + 1.5 ==3 u^2

The area of the vertical rectangle ==2  x  4 ==8 u^2

The shaded area ==area of the vertical rectangle - the sum of 2 triangles at the top and the bottom

== 8  -  3 == 5 u^2 - area the shaded section.

Guest Nov 30, 2022
#3
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I admit the last part was wrong, it is in fact 5

Dec 1, 2022
edited by Imcool  Dec 1, 2022
#4
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I got a bit too excited, i was 2x2.5 not 2.5x2.5 im sorry

Dec 1, 2022