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# Geometry

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In the diagram, the area of rectangle EFGH is $$1/5$$ of the area of unit square ABCD. What is the area of quadrilateral AFCH in square units?

May 8, 2022

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So we know that a unit square is just a 1 by 1 square, so the area of a unit square is 1.

Since rectangle EFGH has 1/5 of the area of the unit square, the area of EFGH is 1/5.

For simplicity, area will be denoted by [] below. (For example, the sentence above translates to [EFGH] = 1/5.)

$$[CGF] = \dfrac12 [BCGF]$$ (imagine the rectangle BCGF is a piece of sandwich and you want to cut it in halves.)

Similarly, $$[AEH] = \dfrac12 [ADHE]$$.

Then, we do some algebra.

$$\quad [AFCH]\\ = [CGF] + [EFGH] + [AEH]\\ = [EFGH] + \dfrac12 [BCGF] + \dfrac12 [ADHE]\\ = \dfrac15 + \dfrac12 \left([BCGF] + [ADHE]\right)$$

Then notice that [BCGF] + [ADHE] is just the remaining area when [EFGH] is taken away from the unit square, so that is 4/5.

$$[AFCH] = \dfrac15 + \dfrac12 \cdot \dfrac45 = \text{(you do the remaining calculation)}$$

May 8, 2022