2019 AMC 10B Problems/Problem 23
- The following problem is from both the 2019 AMC 10B #23 and 2019 AMC 12B #20, so both problems redirect to this page.
Points and lie on circle in the plane. Suppose that the tangent lines to at and intersect at a point on the -axis. What is the area of ?
First, observe that the two tangent lines are of identical length. Therefore, supposing that the point of intersection is , the Pythagorean Theorem gives . This simplifies to .
Further, notice (due to the right angles formed by a radius and its tangent line) that the quadrilateral (a kite) defined by the circle's center, , , and is cyclic. Therefore, we can apply Ptolemy's Theorem to give , where is the radius of the circle and is the distance between the circle's center and . Therefore, . Using the Pythagorean Theorem on the triangle formed by the point , either one of or , and the circle's center, we find that , so , and thus the answer is .
Solution 2 (coordinate bash)
We firstly obtain as in Solution 1. Label the point as . The midpoint of segment is . Notice that the center of the circle must lie on the line passing through the points and . Thus, the center of the circle lies on the line .
Line is . Therefore, the slope of the line perpendicular to is , so its equation is .
But notice that this line must pass through and . Hence . So the center of the circle is .
Finally, the distance between the center, , and point is . Thus the area of the circle is .
The midpoint of is . Let the tangent lines at and intersect at on the -axis. Then is the perpendicular bisector of . Let the center of the circle be . Then is similar to , so . The slope of is , so the slope of is . Hence, the equation of is . Letting , we have , so .
Now, we compute , , and .
Therefore , and consequently, the area of the circle is .
Solution 4 (how fast can you multiply two-digit numbers?)
Let be the intersection on the x-axis. By Power of a Point Theorem, . Then the equations are and for the tangent lines passing and respectively. Then the lines normal to them are and . Thus,
After condensing, . Then, the center of is . Apply distance formula. WLOG, assume you use . Then, the area of is . ~ minor LaTeX edits by dolphin7
Solution 5 (power of a point)
Firstly, the point of intersection of the two tangent lines has an equal distance to points and due to power of a point theorem. This means we can easily find the point, which is . Label this point . is an isosceles triangle with lengths, , , and . Label the midpoint of segment as . The height of this triangle, or , is . Since bisects , contains the diameter of circle . Let the two points on circle where intersects be and with being the shorter of the two. Now let be and be . By Power of a Point on and , . Applying Power of a Point again on and , . Expanding while using the fact that , . Plugging this into , . Using the quadratic formula, , and since , . Since this is the diameter, the radius of circle is , and so the area of circle is .
For those who want a video solution: (Is similar to Solution 1) https://youtu.be/WI2NVuIp1Ik
|2019 AMC 10B (Problems • Answer Key • Resources)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25|
|All AMC 10 Problems and Solutions|
|2019 AMC 12B (Problems • Answer Key • Resources)|
|All AMC 12 Problems and Solutions|