2017 AMC 12B Problems/Problem 18
The diameter of a circle of radius is extended to a point outside the circle so that . Point is chosen so that and line is perpendicular to line . Segment intersects the circle at a point between and . What is the area of ?
Let be the center of the circle. Note that . However, by Power of a Point, , so . Now . Since .
Solution 2: Similar triangles with Pythagorean
is the diameter of the circle, so is a right angle, and therefore by AA similarity, .
Because of this, , so .
Likewise, , so .
Thus the area of .
Solution 2b: Area shortcut
Because is and is , the ratio of the sides is , meaning the ratio of the areas is thus . We then have the proportion
Solution 3: Similar triangles without Pythagorean
Or, use similar triangles all the way, dispense with Pythagorean, and go for minimal calculation:
Draw with on . .
. ( ratio applied twice)
Solution 4: Coordinate geometry
Let be at the origin of a coordinate plane, with being located at , etc.
We can find the area of by finding the the altitude from line to point . Realize that this altitude is the coordinate of point on the coordinate plane, since the respective base of is on the -axis.
Using the diagram in solution one, the equation for circle is .
The equation for line is then , therefore .
Substituting for in the equation for circle , we get:
We can solve for to yield the coordinate of point in the coordinate plane, since this is the point of intersection of the circle and line . Note that one root will yield the intersection of the circle and line at the origin, so we will ignore this root.
Expanding the expression and factoring, we get:
Our non-zero root is thus . Calculating the area of with as the length of and as the altitude, we get:
-Solution by Joeya
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