Question

Find the ratio of the areas of quadrant of a circle to area of semicircle of radius of same radius

Answer 1

Step-by-step explanation:

Without loss of generality, we can let the radius of the semi-circle be 1 and the radius of the quarter-circle be r. Then, the desired ratio R is given by

R = (pi/2)*r²/(pi/4) = 2*r²

Now we need to calculate r.

Since the diameter (of length 2*r) of the quarter-circle is also a chord of the semi-circle, we know from geometry that

2 = r²/x + x

where x is the distance along the quarter-circle's radius from the edge of the quarter-circle to the chord. A bit of geometry tells us that

x = 1 - r*sqrt(2)

Inserting this into the previous equation and solving for r² gives

r² = 1/3

Finally, plugging this into the R equation gives the result

R = 2/3

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Answer 2

Step-by-step explanation:

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