a semicircle is inscribed in a quarter circle as illustrated. what fraction of the quarter circle is shaded?
Answers
The vertical and horizontal radii of the semicircle are tangent to the quarter circle, so they form right angles. As a quarter circle also spans 90 degrees, we have a quadrilateral with 3 right angles and two sides equal to s. Thus the figure is a square, so all sides are equal to s and its diagonal is s√2.)
Now we will draw one radius of the quarter circle through the center of the semicircle. This will bisect the semicircle’s diameter–a chord of the quarter circle–so the two line segments meet at a right angle. We also draw a radius from the quarter circle to where the diameter of the semicircle meets the quarter circle.
Now we have a right triangle with legs s√2 and s, and a hypotenuse q, so we can use the Pythagorean Theorem to get:
(s√2)2 + s2 = q2
3s2 = q2
That’s the equation we need to solve this problem. We now go back to what we derived earlier:
(area semicircle)/(area quarter circle)
= (0.5πs2)/(0.25πq2)
= 2s2/q2
We substitute 3s2 = q2 to get:
(area semicircle)/(area quarter circle)
= 2s2/(3s2)
= 2/3
So the fraction that is shaded is 2/3.
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