2003 AMC 12A Problems/Problem 17
Square has sides of length , and is the midpoint of . A circle with radius and center intersects a circle with radius and center at points and . What is the distance from to ?
Let be the origin. is the point and is the point . We are given the radius of the quarter circle and semicircle as and , respectively, so their equations, respectively, are:
Subtract the second equation from the first:
Thus and making and .
The first value of is obviously referring to the x-coordinate of the point where the circles intersect at the origin, , so the second value must be referring to the x coordinate of . Since is the y-axis, the distance to it from is the same as the x-value of the coordinate of , so the distance from to is
obviously forms a kite. Let the intersection of the diagonals be . Let . Then, .
By Pythagorean Theorem, . Thus, . Simplifying, . By Pythagoras again, . Then, the area of is .
Using instead as the base, we can drop a altitude from P. . . Thus, the horizontal distance is
Note that is merely a reflection of over . Call the intersection of and . Drop perpendiculars from and to , and denote their respective points of intersection by and . We then have , with a scale factor of 2. Thus, we can find and double it to get our answer. With some analytical geometry, we find that , implying that .
As in Solution 2, draw in and and denote their intersection point . Next, drop a perpendicular from to and denote the foot as . as they are both radii and similarly so is a kite and by a well-known theorem.
Pythagorean theorem gives us . Clearly by angle-angle and by Hypotenuse Leg. Manipulating similar triangles gives us
Using the double-angle formula for sine, what we need to find is .
We use the Law of Cosines:
Let be the foot of the perpendicular from to , and let . Then we have , and . Since , we have , or . Solving gives .
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