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Let $ABCD$ be a right trapezoid,and $AB$ is diameter, and where

$AB=2R$,

$DE:EF:CF=12:15:5$.

Find the $S_{ABCD}$

enter image description here

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    What is the source of the problem? Can you let us know what you've already done to solve the problem. And can you explain what you mean by $S_{ABCD}$? – 2017-01-03
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    What does $S_{ABCD}$ mean? The perimeter? – 2017-01-03

3 Answers 3

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$$S(ABCD)=\frac{2R\cdot(AD+BC)}{2}=R\cdot(AD+BC)$$

Let's write $DE=12k$, $EF=15k$ and $CF=5k$

Using power of the point $D$ we have

$$AD^2=DE\cdot DF \rightarrow AD^2=12k\cdot 27k \rightarrow AD=18k$$

and by the power of the point $C$ we get

$$BC^2=CF\cdot CE \rightarrow BC^2=5k \cdot 20k \rightarrow BC=10k$$ enter image description here

And looking the triangle $CDE$ we know that $AD-BC=8k$ and by Pythagoras theorem:

$$(32k)^2=(2R)^2+(8k)^2\rightarrow k=\frac{R}{4\sqrt{15}}$$

And then

$$S(ABCD)=R\cdot 28\cdot \frac{R}{4\sqrt{15}}=\frac{7R^2}{\sqrt{15}}$$

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Hint: $AD$ is a tangent to the circle, so $AD^2 = DE\cdot DF$. Similarly for the tangent from $C$.

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Observe that $BC^2=CF\times CE$ and $AD^2=DE\times DF$