Christmas Streak 03/88: Three Lines + One Circle
Image not drawn to scale.
Tom has drawn a circle and three straight line segments.
What is the value of in the picture shown, to three decimal places?
This problem is a part of <Christmas Streak 2017> series .
The answer is 12.000.
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Extend the line segment a little bit more, and you see the picture above.
Using the power theorem , we figure out that
x y = 8 ⋅ 9 ; ( 1 6 + y ) ( 1 6 − x ) = 8 ⋅ 1 1 .
Since y = x 7 2 ,
( 1 6 + x 7 2 ) ( 1 6 − x ) = 8 8 8 ( 2 + x 9 ) ( 1 6 − x ) = 8 8 2 3 + x 1 4 4 − 2 x = 1 1 2 x − x 1 4 4 − 1 2 = 0 x 2 − 6 x − 7 2 = 0 ( x − 1 2 ) ( x + 6 ) = 0 ∴ x = 1 2 .