AMC 10 · 2013 · #23

Grade 7 geometry-2d
Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration
power-of-a-pointisosceles-trianglepolygon-inequality convert-to-algebrabound-inequality-then-enumerate ↑ Prerequisites: power-of-a-point
📏 Long solution 💡 3 insights
Problem
In triangle ABC the side AB is 86 and the side AC is 97. A circle is centered at A with radius 86, the same length as AB. This circle crosses side BC at B and at one more point X. The two pieces of the side, BX and CX, both turn out to be whole numbers. Find the length of the whole side BC.

Pick an answer.

(A)
11
(B)
28
(C)
33
(D)
61
(E)
72

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The two side pieces are hidden, so name them with letters. A circle through B and X lets me turn the side lengths into one clean product using the Power of a Point idea. That product is a fixed number, so I factor it to list the possible values of BC, then use the triangle inequality to erase every value but one.

1STEP 1

Draw the circle and mark radii

Sketch the circle of radius 86 about A, so AX = AB = 86. Since AC = 97, line CA meets it 11 and 183 from C.

AX = AB = 86, C outside since AC = 97 ＞ 86, line CA meets circle at distances 97-86=11 and 97+86=183 from C
2STEP 2

Name the two side pieces

Let CX = x and BX = y be positive whole numbers, so BC = x + y, which is longer than CX = x.

CX = x, BX = y, BC = x + y, CB ＞ CX
3STEP 3

Use Power of a Point at C

Two lines from C cut the circle: CB at x and x + y, CA at 11 and 183. Power of a Point gives x(x + y) = 2013.

CX · CB = x(x+y) = 11 · 183 = 2013
4STEP 4

Factor 2013 to list the options

Factor: 2013 = 3 · 11 · 61, so the pair (x, x + y) is (1, 2013), (3, 671), (11, 183), or (33, 61).

2013 = 3 · 11 · 61, (x, x+y) ∈ {(1,2013), (3,671), (11,183), (33,61)}
5STEP 5

Erase impossible lengths

The triangle inequality needs BC between 11 and 183, so only BC = 61 fits, with CX = 33 and BX = 28.

11 ＜ BC ＜ 183 → BC = 61, CX = 33, BX = 28
Answer
61
Test BC = 61. The pieces are CX = 33 and BX = 28, which add to 61 and are both whole numbers. The product 33 * 61 = 2013 matches 11 * 183, so the Power of a Point equation holds. The three sides 86, 97, 61 obey the triangle inequality, since 61 ＜ 86 + 97, 86 ＜ 61 + 97, and 97 ＜ 61 + 86. Everything checks, so (D) is solid.
💡Key takeaway

From one point outside a circle, every straight cut gives the same product of distances, so name the pieces, get one product, and factor it.

  • Draw the circle and mark radii
  • Name the two side pieces
  • Use Power of a Point at C
  • Factor 2013 to list the options
  • Erase impossible lengths

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