Try this beautiful problem from Geometry: Area of the region

Problem on Area of the Region – AMC-10A, 2007- Problem 24


Circle centered at \(A\) and \(B\) each have radius \(2\), as shown. Point \(O\) is the midpoint of \(\overline{AB}\), and \(OA = 2\sqrt {2}\). Segments \(OC\) and \(OD\) are tangent to the circles centered at $A$ and $B$, respectively, and $EF$ is a common tangent. What is the area of the shaded region \(ECODF\)?

  • \(\pi\)
  • \(7\sqrt 3 -\pi\)
  • \(8\sqrt 2 -4-\pi\)

Key Concepts


Geometry

Triangle

similarity

Check the Answer


But try the problem first…

Answer: \(8\sqrt 2 -4-\pi\)

Source
Suggested Reading

AMC-10A (2007) Problem 24

Pre College Mathematics

Try with Hints


First hint

Area of the region problem

We have to find out the area of the region \(ECODF\) i.e of gray shaded region.this is not any standard geometrical figure (such as circle,triangle…etc).so we can not find out the value easily.Now if we join \(AC\),\(AE\),\(BD\),\(BF\).Then \(ABFE\) is a rectangle.then we can find out the required area by [ area of rectangle \(ABEF\)- (area of arc \(AEC\)+area of \(\triangle ACO\)+area of \(\triangle BDO\)+ area of arc \(BFD\))]

Shaded region

Can you find out the required area…..?

Second Hint

Given that Circle centered at \(A\) and \(B\) each have radius \(2\) and Point \(O\) is the midpoint of \(\overline{AB}\), and \(OA = 2\sqrt {2}\)

Shaded area of the region

Area of \(ABEF\)=\(2 \times 2 \times 2\sqrt 2\)=\(8\sqrt 2\)

Now \(\triangle{ACO}\) is a right triangle. We know \(AO=2\sqrt{2}\)and \(AC=2\), so \(\triangle{ACO}\) is isosceles, a \(45\)-\(45\) right triangle.\(\overline{CO}\) with length \(2\). The area of \(\triangle{ACO}=\frac{1}{2} \times base \times height=2\). By symmetry, \(\triangle{ACO}\cong\triangle{BDO}\), and so the area of \(\triangle{BDO}\) is also \(2\).now the \(\angle CAO\) = \(\angle DBO\)=\(45^{\circ}\). therefore \(\frac{360}{45}=8\)

So the area of arc \(AEC \) and arc \(BFD\)=\(\frac{1}{8} \times\) area of the circle=\(\frac{\pi 2^2}{8}\)=\(\frac{\pi}{2}\)

can you finish the problem……..

Final Step

Therefore the required area by [ area of rectangle \(ABEF\)- (area of arc \(AEC\)+area of \(\triangle ACO\)+area of \(\triangle BDO\)+ area of arc \(BFD\))]=\(8\sqrt 2-(\frac{\pi}{2}+2+2+\frac{\pi}{2}\))=\(8\sqrt 2 -4-\pi\)

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