MO circles

Juro built the ABCD square with a 12 cm side. In this square, he scattered a quarter circle that had a center at point B passing through point A and a semicircle l that had a center at the center of the BC side and passed point B. He would still build a circle that would lie inside the square and touch the quarter circle k, semicircle l and side AB. Find the radius of such circle.

Result

r =  3 cm

Solution:

SO=SL+LO=R+6 SB=BKKS=12r OE=OBBE=6r  SE²=SO²OE²=SB²BE² (6+r)²(6r)²=(12r)²r² 12r+12r=14424r 48r=144 r=144/48=3=3  cm |SO| = |SL| + |LO| = R + 6 \ \\ |SB| = |BK | - |KS| = 12 - r \ \\ |OE| = |OB| - |BE| = 6 - r \ \\ \ \\ |SE|² = |SO|² - |OE|² = |SB|² - |BE|² \ \\ (6 + r)² - (6 - r)² = (12 - r)² - r² \ \\ 12r + 12r = 144 - 24r \ \\ 48r = 144 \ \\ r = 144/48 = 3 = 3 \ \text{ cm }



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