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In rectangle ABCD, point E lies half way between sides AB and CD and halfway between sides AD and BC. If AB=11 and BC=6, what is the area of the shaded region? Write your answer as a decimal, if necessary. Do not include units in your answer.

User Kierchon
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18 votes

Answer:

The area of the shaded region is 36.

In rectangle ABCD, point E lies halfway between sides AB and CD and halfway between sides AD and BC. This means that line segment AE is parallel to line segment CD and is half as long, and line segment BE is parallel to line segment AD and is also half as long.

Since AE is half as long as CD, and CD is twice as long as AE, we can say that CD is equal to 2 * AE. Similarly, since BE is half as long as AD, and AD is twice as long as BE, we can say that AD is equal to 2 * BE.

Since AD is equal to 2 * BE and CD is equal to 2 * AE, the lengths of AD and CD are equal. This means that the shaded region is a parallelogram with sides of equal length.

The area of a parallelogram is equal to the product of its base and height. In this case, the base of the parallelogram is 6 (the length of side BC) and the height is AE + BE (the combined length of the two sides of the parallelogram).

We know that AE is half as long as CD, and CD is twice as long as AE, so CD is equal to 2 * AE. Since CD is equal to 2 * AE, AE is equal to CD / 2. Since CD is equal to 11 (the length of side AB), AE is equal to 11 / 2 = 5.5.

Similarly, we know that BE is half as long as AD, and AD is twice as long as BE, so AD is equal to 2 * BE. Since AD is equal to 2 * BE, BE is equal to AD / 2. Since AD is equal to 6 (the length of side BC), BE is equal to 6 / 2 = 3.

The combined length of AE and BE is 5.5 + 3

User Godsmith
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