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Try this beautiful problem from Geometry based on Semicircle.

A $1\times 2$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?

- \( \frac{\pi}{2}\)
- \(\pi\)
- \( \frac{\pi}{3}\)

Geometry

Square

semi circle

But try the problem first...

Answer:\(\pi\)

Source

Suggested Reading

AMC-8 (2013) Problem 20

Pre College Mathematics

First hint

At first we have to find out the radius of the semicircle for the area of the semicircle.Now in the diagram,AC is the radius of the semicircle and also AC is the hypotenuse of the right Triangle ABC.

Can you now finish the problem ..........

Second Hint

Now AB=1 AND AC=1 (As ABDE $1\times 2$ rectangle .So using pythagorean theorm we can eassily get the value of AC .and area of semicircle =\(\frac{\pi r^2}{2}\)

can you finish the problem........

Final Step

Given that ABDE is a square whose AB=1 and BD=2

Therefore BC=1

Clearly AC be the radius of the given semi circle

From the \(\triangle ABC\),\((AB)^2+(BC)^2=(AC)^2\Rightarrow AC=\sqrt{(1^2+1^2)}=\sqrt2\)

Therefore the area of the semicircle=\(\frac{1}{2}\times \pi(\sqrt 2)^2\)=\(\pi\)

- https://www.cheenta.com/pattern-problem-amc-8-2002-problem-23/
- https://www.youtube.com/watch?v=qb4oN02rSxM

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