2005 AMC 10A Problems/Problem 23
Let be a diameter of a circle and let be a point on with . Let and be points on the circle such that and is a second diameter. What is the ratio of the area of to the area of ?
Let us assume that the diameter is of length .
is of diameter and is .
is the radius of the circle, so using the Pythagorean theorem height of is . This is also the height of the .
Area of the is = .
The height of can be found using the area of and as base.
Hence the height of is = .
The diameter is the base for both the triangles and .
Hence, the ratio of the area of to the area of is =
Since and share a base, the ratio of their areas is the ratio of their altitudes. Draw the altitude from to .
Since , then . So the ratio of the two altitudes is
Say the center of the circle is point ; Without loss of generality, assume , so and the diameter and radius are and , respectively. Therefore, , and . The area of can be expressed as happens to be the area of . Furthermore, or Therefore, the ratio is
We know that and is a diameter. Without loss of generality, assume the diameter has length 6, which means that and .
WLOG, let , , so radius of the circle is and . As in solution 1, By same altitude, the ratio , where is the point where extended meets circle . Note that angle P = 90 deg, so DCO ~ DPE with ratio 1:2, so PE = 1. Thus, our ratio is .
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