2017 AMC 12B Problem 6

Below is the professionally curated solution for Problem 6 of the 2017 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12B solutions, or check the answer key.

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Concepts:coordinate geometry circle midpoint

Difficulty rating: 1350

The circle having $(0, 0)$ and $(8, 6)$ as the endpoints of a diameter intersects the $x$ -axis at a second point. What is the $x$ -coordinate of this point?

$4\sqrt{2}$

$6$

$5\sqrt{2}$

$8$

$6\sqrt{2}$

Solution:

The center is the midpoint of the diameter, $(4, 3),$ and the radius is $\sqrt{4^2 + 3^2} = 5.$ The circle is $(x - 4)^2 + (y - 3)^2 = 25.$ Setting $y = 0$ gives $(x - 4)^2 = 16,$ so $x = 0$ or $x = 8.$ The second intersection with the $x$ -axis is at $x = 8.$

Thus, the correct answer is D.

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