Hard kick

Geometry Level pending

During hard training, a martial artist applied a hard semi-circular low kick that breaks a vertical wooden pole 6 6 feet high at point B B as shown above. The pole remain in contact at B B and the top of the pole touches the horizontal ground at X X , a distance 3 1 2 3\frac{1}{2} feet from point A A . Find the length A B AB in feet.

115 58 \dfrac{115}{58} 95 48 \dfrac{95}{48} 20 11 \dfrac{20}{11} 57 29 \dfrac{57}{29}

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Consider the diagram on the left. Let A B = y AB=y , then B X = 6 − y BX=6-y .

By pythagorean theorem, we have

( 6 − y ) 2 = y 2 + ( 7 2 ) 2 (6-y)^2=y^2+\left(\dfrac{7}{2}\right)^2

36 − 12 y + y 2 = y 2 + 49 4 36-12y+y^2=y^2+\dfrac{49}{4}

12 y = 36 − 49 4 = 144 − 49 4 = 95 4 12y=36-\dfrac{49}{4}=\dfrac{144-49}{4}=\dfrac{95}{4}

y = 95 12 ( 4 ) = y=\dfrac{95}{12(4)}= 95 48 \boxed{\dfrac{95}{48}}

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