AMC 10 · 2016 · #19

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Vertices then Area

coordinate-geometryslope-interceptpythagorean-theorem convert-to-algebra ↑ Prerequisites: coordinate-geometry

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A 5-by-4 rectangle ABCD has three marked points: E on AB with EB=1, G on BC with CG=1, and F on CD with DF=2. The diagonal AC and the segment AG both cross the segment EF, at points P and Q. Find what fraction of EF the piece PQ takes up.

Pick an answer.

(A)

$~\frac{\sqrt{13}}{16}$

(B)

$~\frac{\sqrt{2}}{13}$

(C)

$~\frac{9}{82}$

(D)

$~\frac{10}{91}$

(E)

$~\frac19$

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The figure is full of named points and crossing segments, so the cleanest move is to drop the whole rectangle onto a coordinate grid. Once every point has numbers, each line becomes an equation, and finding where two segments cross becomes solving two equations together. The big question PQ/EF then splits into small subproblems: locate P, locate Q, then measure.

1STEP 1

Put the rectangle on a grid

Put D at the origin, so A=(0,4) and C=(5,0); the three marked points become E=(4,4), F=(2,0), G=(5,1).

A=(0,4), B=(5,4), C=(5,0), D=(0,0), E=(4,4), G=(5,1), F=(2,0)

2STEP 2

Write the line through E and F

From F=(2,0) to E=(4,4) the slope is 2, so line EF is y=2x-4; every point of EF, P and Q included, satisfies it.

slope_EF=(4-0)/(4-2)=2, y=2x-4

3STEP 3

Find P where AC meets EF

Line AC is y=4-(4/5)x; set it equal to y=2x-4 to get x=20/7, so P=(20/7, 12/7).

4-4/5x=2x-4 → 14/5x=8 → x=20/7, P=(20/7,12/7)

4STEP 4

Find Q where AG meets EF

Line AG is y=4-(3/5)x; set it equal to y=2x-4 to get x=40/13, so Q=(40/13, 28/13).

4-3/5x=2x-4 → 13/5x=8 → x=40/13, Q=(40/13,28/13)

5STEP 5

Measure PQ and EF, then divide

PQ=(20/91)√5 and EF=2√5 lie on one line, so the √5 cancels: PQ/EF=(20/91)√5 ÷ 2√5=10/91.

PQ/EF=20/91√5/2√5=10/91

Answer

~10/91

Numerically PQ≈0.49 and EF≈4.47, giving a ratio near 0.11, which matches 10/91≈0.110. PQ is a small slice in the middle of EF, so a small fraction below 1/8 is sensible. The denominator 91=7·13 is exactly the product of the two denominators that appeared when locating P (7) and Q (13), a reassuring sign no arithmetic slipped.

💡Key takeaway

Put the whole picture on a grid, turn each line into an equation, and the crossing points are just where two equations agree.

Put the rectangle on a grid
Write the line through E and F
Find P where AC meets EF
Find Q where AG meets EF
Measure PQ and EF, then divide

Put the rectangle on a grid

Write the line through E and F

Find P where AC meets EF

Find Q where AG meets EF

Measure PQ and EF, then divide

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