Grade 11 Math Puzzle 12: Coordinate normalization

A free grade 11 math puzzle that teaches Coordinate normalization. Try it first — the hint and the full solution open when you are ready.

Problem 12 · Coordinate normalization · 4 points

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On a triangular mural ABC, P lies on AB with AP:PB = 2:1, and Q lies on AC with AQ:QC = 1:2. If CP and BQ cross at X, what fraction of the mural is triangle BXC?

triangle ABC with P on side AB two thirds of the way from A to B (AP:PB = 2:1), Q on side AC one third of the way from A to C (AQ:QC = 1:2), the segments CP and BQ crossing at X, the points A, B, C, P, Q and X labelled, and triangle BXC shaded

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Use AB and AC as coordinate directions; they need not be perpendicular.
Find X, then compare areas with the whole triangle.

Show the answer and solution

Answer: C — 2/7

Use coordinates A = (0,0), B = (1,0), and C = (0,1); this change of shape preserves area ratios. Then P = (2/3,0) and Q = (0,1/3). The lines CP and BQ have equations 3x + 2y = 2 and x + 3y = 1, so X = (4/7,1/7). Triangles ACX and ABX occupy 4/7 and 1/7 of ABC, respectively. The remaining triangle BXC therefore occupies 1 − 4/7 − 1/7 = 2/7. The tempting choice 5/7 adds the two triangles beside A instead of finding the requested triangle.

Grade 11 puzzle 12: Coordinate normalization

Problem 12 · Coordinate normalization · 4 points