2017 AMC 12B Problems/Problem 6
Problem 6
The circle having and as the endpoints of a diameter intersects the -axis at a second point. What is the -coordinate of this point?
Solution 1
Because the two points are on a diameter, the center must be halfway between them at the point (4,3). The distance from (0,0) to (4,3) is 5 so the circle has radius 5. Thus, the equation of the circle is .
To find the x-intercept, y must be 0, so , so , , .
Written by: SilverLion
Solution 2 (no calculation)
As in solution 1, we find that the midpoint of the circle is (4,3) by finding half of the x and y coordinates of (8,6) point. We see a 3-4-5 triangle! For a point to be on the circle, it has to be 5 units away from the center.
We are able to create another 3-4-5 triangle by drawing a segment from (4,3) to (0,8). Therefore we know our answer is
https://www.geogebra.org/geometry/xryzxpyw
-thedodecagon
See Also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
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All AMC 12 Problems and Solutions |
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