2021 AMC 12A Problems/Problem 24
- 1 Problem
- 2 Diagram
- 3 Solution 1 (Possible Without Trigonometry)
- 4 Solution 2 (Trigonometry)
- 5 Solution 3 (Weighted Averages and Similar Triangles)
- 6 Solution 4 (Similar Triangles)
- 7 Solution 5 (Analytic Geometry)
- 8 Video Solution by MOP 2024
- 9 Video Solution by Punxsutawney Phil
- 10 Video Solution by OmegaLearn (Similar Triangles, Law of Sines, Law of Cosines )
- 11 Video Solution
- 12 See also
Semicircle has diameter of length . Circle lies tangent to at a point and intersects at points and . If and , then the area of equals , where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. What is ?
Solution 1 (Possible Without Trigonometry)
Let be the center of the semicircle and be the center of the circle.
Applying the Extended Law of Sines to we find the radius of Alternatively, by the Inscribed Angle Theorem, is a triangle with base Dividing into two congruent triangles, we get that the radius of is by the side-length ratios.
Let be the midpoint of By the Perpendicular Chord Bisector Converse, we have and Together, points and must be collinear.
By the SAS Congruence, we have both of which are triangles. By the side-length ratios, we obtain and By the Pythagorean Theorem on right we get and By the Pythagorean Theorem on right we get
Let be the foot of the perpendicular from to and be the foot of the perpendicular from to as shown below: Clearly, quadrilateral is a rectangle. Since by alternate interior angles, we have by the AA Similarity, with the ratio of similitude Therefore, we get and
The area of is from which the answer is
Solution 2 (Trigonometry)
Suppose we label the points as shown in the diagram above, where is the center of the semicircle and is the center of the circle tangent to . Since , we have and is a triangle, which can be split into two triangles by the altitude from . Since we know by triangles. The area of this part of is . We would like to add this value to the sum of the areas of the other two parts of .
To find the areas of the other two parts of using the area formula, we need the sides and included angles. Here we know the sides but what we don't know are the angles. So it seems like we will have to use an angle from another triangle and combine them with the angles we already know to find these angles easily. We know that and triangles and are congruent as they share a side, and . Therefore . Suppose . Then , and since , this simplifies to . This factors nicely as , so as can't be . Since and , we now know that is a right triangle. This may be useful info for later as we might use an angle in this triangle to find the areas of the other two parts of .
Let . Then and . The sum of the areas of and is which we will add to to get the area of . Observe that and similarly . Adding these two gives and multiplying that by gets us which we add to to get . The answer is
Solution 3 (Weighted Averages and Similar Triangles)
Define points as shown above, where . The area of is simply it remains to compute the value of . Note that is simply a weighted average of and it is times closer to than it is to . Observe that since the radius of is as its diameter is . Note also by the Extended Law of Sines the radius of is so . Since and are collinear by symmetry we have so and . Therefore, is a right triangle; since and . Therefore so . Since we have . Therefore so is times as close to as to we can compute . The area of isand .
Solution 4 (Similar Triangles)
Let be the center of be the center of and be the midpoint of We have and by Extended Law of Sines, the radius of is so Therefore and
Let Obviously so with ratio Therefore and
Let denote the foot of the altitude from to Because it follows that This similarity has ratio We therefore have
Finally, the area of is so the answer is
Solution 5 (Analytic Geometry)
Following Solution 4, We have (0,0) , (4,3). We can write the equation of the two circles as: By substituting (1) into (2), we get Notice (3) is the relationship between value and value, in other words, (3) is the linear equation that go through and . Let the height drops from to at . Therefore, we have So And by distance formula, is the distance from (4,0) to . Thus, We get So the answer is
Video Solution by MOP 2024
Video Solution by Punxsutawney Phil
This is invalid. ~r00tsOfUnity
Video Solution by OmegaLearn (Similar Triangles, Law of Sines, Law of Cosines )
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