2017 AMC 10B Problem 8

Below is the professionally curated solution for Problem 8 of the 2017 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 10B solutions, or check the answer key.

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Concepts:coordinate geometry midpoint isosceles triangle

Difficulty rating: 1070

Points $A(11, 9)$ and $B(2, -3)$ are vertices of $\triangle ABC$ with $AB=AC.$ The altitude from $A$ meets the opposite side at $D(-1, 3).$ What are the coordinates of point $C?$

$(-8, 9)$

$(-4, 8)$

$(-4, 9)$

$(-2, 3)$

$(-1, 0)$

Solution:

Since the triangle $ABC$ is isosceles, the altitude from $A$ is the midpoint of the other two sides. Therefore, $D$ is the midpoint between $B$ and $C.$ If $C=(x,y),$ then we have $\frac{x+2}2 = -1,$ $\frac{y-3}2=3.$ As such, $C=(x,y)=(-4,9).$

Thus, the correct answer is C .

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