AMC 8 · 2002 · #1

Grade 7 geometry-2dcounting

Archetype Arithmetic Expression Decomposition

spatial-visualizationsystematic-enumerationcombinations-basic identify-subproblemssystematic-enumeration ↑ Prerequisites: systematic-enumeration

📏 Short solution 💡 2 insights

Problem

A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures?

$\text {(A)}\ 2 \qquad \text {(B)}\ 3 \qquad {(C)}\ 4 \qquad {(D)}\ 5 \qquad {(E)}\ 6$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: A circle and two distinct straight lines are drawn on the same sheet of paper. Counting every point where any two of these three figures cross, what is the largest possible total number of intersection points?

Givens: Three figures share one plane: $1$ circle and $2$ distinct lines; We want the maximum possible number of intersection points; Answer choices: (A) $2$, (B) $3$, (C) $4$, (D) $5$, (E) $6$

Unknowns: The largest possible total count of intersection points

Understand

Givens: Three figures share one plane: $1$ circle and $2$ distinct lines; We want the maximum possible number of intersection points; Answer choices: (A) $2$, (B) $3$, (C) $4$, (D) $5$, (E) $6$

Plan

Primary tool: #7 Break into Subproblems

Secondary: #15 Visualize

Three figures make three pairs: line-line, line-circle (twice). Tool #7 (Break into Subproblems) lets us handle one pair at a time, find the maximum for each, and add. Tool #15 (Visualize) is needed at the end to draw a single picture where every pair hits its maximum at the same time, so the five points are all different and the total is actually reachable.

Execute — Answer: D

#7 Break into Subproblems 4.G.A.1 Step 1

Pair 1: the two lines.
Two distinct straight lines either miss each other (parallel) or cross at exactly one point.
To make the count as big as possible, let them cross.

$$\text{line} \cap \text{line} \le 1 \text{ point}$$

💡 Grade 4 introduces points, lines, and parallel vs. intersecting lines: two different lines share at most one point.

#7 Break into Subproblems 7.G.B.4 Step 2

Pair 2: the circle and the first line.
A line can miss a circle ($0$ points), touch it as a tangent ($1$ point), or cut through it as a secant ($2$ points).
To maximize, send the line through the inside of the circle.

$$\text{circle} \cap \text{line}_1 \le 2 \text{ points}$$

💡 Grade 7 work with circles makes the secant case familiar: a chord meets the circle at its two endpoints.

#7 Break into Subproblems 7.G.B.4 Step 3

Pair 3: the circle and the second line.
Same situation, same maximum.

$$\text{circle} \cap \text{line}_2 \le 2 \text{ points}$$

💡 The second line is independent of the first, so it can also be a secant and contribute $2$ more points.

#7 Break into Subproblems 4.OA.A.3 Step 4

Add the three pair-maximums to get an upper bound on the total.
This is the largest the count could possibly be.

$$1 + 2 + 2 = 5$$

💡 Grade 4 multi-step addition: each pair contributes independently, so the totals just stack up.

#15 Visualize 4.G.A.1 Step 5

Check the bound is actually reachable.
Draw a circle, then two secant lines that cross each other inside the circle.
The two lines give $1$ intersection inside; each line cuts the circle at $2$ fresh points, for $4$ more.
All $5$ points are different, so the maximum is achieved.

$$\text{achievable total} = 5 \;\Rightarrow\; \textbf{(D)}$$

💡 Grade 4 "draw and identify" geometry: a quick sketch confirms the five points are distinct, so the upper bound $5$ is actually attained.

[1] #7 4.G.A.1 Pair 1: the two lines. Two distinct straight lines either miss each other (paral

[2] #7 7.G.B.4 Pair 2: the circle and the first line. A line can miss a circle ($0$ points), to

[3] #7 7.G.B.4 Pair 3: the circle and the second line. Same situation, same maximum.

[4] #7 4.OA.A.3 Add the three pair-maximums to get an upper bound on the total. This is the larg

[5] #15 4.G.A.1 Check the bound is actually reachable. Draw a circle, then two secant lines that

Review

Reasonableness: Counting cross-checks: with $3$ figures there are $\binom{3}{2} = 3$ pairs, and the per-pair maxes are $1$, $2$, $2$, which sum to $5$ — matching choice (D). Pushing further is impossible: a third intersection on either line-circle pair would force a line to meet a circle in $3$ points (no such line exists), and a second line-line intersection would force the two lines to coincide (not distinct). So $5$ is both an upper bound and reachable.

Alternative: Tool #13 (Count Strategically): use $\binom{3}{2}=3$ pairs and the per-pair caps directly. Two lines: cap $1$. Line and circle: cap $2$. Apply caps $1 + 2 + 2 = 5$, then sketch one configuration that realizes the cap. Same answer (D), reached by counting the pairs first instead of walking through them one by one.

CCSS standards used (min grade 7)

4.G.A.1 Draw points, lines, line segments, rays, angles, and perpendicular and parallel lines. Identify these in two-dimensional figures (Knowing two distinct lines meet in at most one point, and sketching the configuration that realizes five distinct intersection points.)
7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between circumference and area (Working with a circle and its secants — the line-circle intersection count of at most $2$ comes from Grade 7 circle reasoning.)
4.OA.A.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations (Adding the three pair-maximums $1 + 2 + 2 = 5$ to get the overall bound.)

⭐ Three figures make three pairs. Find each pair's biggest intersection count, add them up, and then make sure one picture can hit all the maximums at once — that final check is what turns $5$ from a guess into the real answer.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 7)

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