2013 AMC 12A Problem 19

Problem 19 of 25HarderGeometry Number Theory

In

\triangle ABC,

AB = 86,

and

AC = 97.

A circle with center

A

and radius

AB

intersects

\overline{BC}

at points

B

and

X.

Moreover

\overline{BX}

and

\overline{CX}

have integer lengths. What is

BC?

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Solution

By the Power of a Point Theorem,

BC\cdot CX = AC^2 - AB^2

where

AB

is the radius. Thus

BC\cdot CX = 97^2 - 86^2 = 2013.

Since

BC = BX + CX

and

CX

are integers, they are complementary factors of

2013 = 3\cdot 11\cdot 61.

As

CX \lt BC \lt AB + AC = 183,

the only possibility is

CX = 33

and

BC = 61.

Thus, the correct answer is D.

Tagged: power of a point · prime factorization

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