Calculate area of polygon inside unit square

Let $E(x,y),$ $A(0,0)$, $D(0,1),$ $B(1,0)$ and $C(1,1).$

Thus, we obtain the following system. $$\left(x-\frac{1}{2}\right)^2+(y-1)^2=\left(\frac{1}{2}\right)^2$$ and $$\frac{\sqrt{x^2+(y-1)^2}}{\sqrt{x^2+y^2}}=\frac{4}{5}.$$ Can you end it now?

I got $E\left(\frac{16}{65},\frac{37}{65}\right)$ or $E\left(\frac{16}{17},\frac{13}{17}\right)$ and from here easy to find the area.


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Let $DE=4k$ and $AE=5k$.

Let $F$ be a point on $AE $ such that $DE\perp AF$. $\triangle ADF\cong \triangle DCE$. Therefore, $AF=4k$ and hence $EF=3k$, $DF=k$.

$$(4k)^2+k^2=1$$

$k^2=\dfrac1{17}$ and the required area is $\displaystyle 1^2-\dfrac12(DE)(CE+AF)=1-\dfrac12(4k)(k+4k)=1-10k^2=\frac7{17}$.

Tags:

Geometry

Area

Euclidean Geometry

Systems Of Equations

Circles

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