AMC 8 · 2023 · #7

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation

coordinate-geometrylinear-equations-two-varslope-intercept coordinate-geometryidentify-subproblems ↑ Prerequisites: coordinate-geometryslope-intercept

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A rectangle, with sides parallel to the $x$ -axis and $y$ -axis, has opposite vertices located at $(15, 3)$ and $(16, 5)$ . A line is drawn through points $A(0, 0)$ and $B(3, 1)$ . Another line is drawn through points $C(0, 10)$ and $D(2, 9)$ . How many points on the rectangle lie on at least one of the two lines?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: A small axis-aligned rectangle sits with one corner at $(15, 3)$ and the opposite corner at $(16, 5)$. One line is drawn through $A(0,0)$ and $B(3,1)$; another line is drawn through $C(0,10)$ and $D(2,9)$. We want to count how many points of the rectangle (its four sides and corners) lie on at least one of those two lines.

Givens: Rectangle has opposite vertices $(15, 3)$ and $(16, 5)$, with sides parallel to the axes; So the rectangle occupies $15 \le x \le 16$ and $3 \le y \le 5$; Line $\ell_1$ passes through $A(0,0)$ and $B(3,1)$; Line $\ell_2$ passes through $C(0,10)$ and $D(2,9)$; Answer choices: (A) 0, (B) 1, (C) 2, (D) 3, (E) 4

Unknowns: The total number of points on the rectangle that lie on $\ell_1$ or $\ell_2$

Understand

Plan

Primary tool: #1 Draw a Diagram

Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities

The picture is already on a grid, so the cleanest approach is to actually extend each line over to the rectangle and look (Tool #1). Because the two lines are independent, we can handle them as two separate small problems and add the answers (Tool #7): "does $\ell_1$ hit the rectangle?" and "does $\ell_2$ hit the rectangle?". Once we know the line's $y$-value at $x = 15$ and $x = 16$, we just compare with the rectangle's $y$-band $[3, 5]$ — a very small case-check that immediately rules out four of the five choices (Tool #3).

Execute — Answer: B

#1 Draw a Diagram 6.G.A.3 Step 1

Read the rectangle as a window.
Because the sides are parallel to the axes and the opposite corners are $(15,3)$ and $(16,5)$, the rectangle is the set of points with $15 \le x \le 16$ AND $3 \le y \le 5$.
So the rectangle is exactly one unit wide and two units tall, sitting between the vertical lines $x=15$ and $x=16$.

$$\text{Rectangle} = \{(x,y) : 15 \le x \le 16,\ 3 \le y \le 5\}$$

💡 Plotting a rectangle from two opposite corners on the grid is the Grade 6 "polygons in the coordinate plane" idea.

#1 Draw a Diagram 8.F.A.3 Step 2

Find the rule for line $\ell_1$ through $A(0,0)$ and $B(3,1)$.
Going from $A$ to $B$ the line rises $1$ for every $3$ steps right, so the slope is $\tfrac{1}{3}$.
Since it passes through the origin, the rule is $y = \tfrac{1}{3}x$.
(Pattern check: at $x=6$ it gives $y=2$, at $x=9$ it gives $y=3$ — yes, $1$ up for every $3$ across.)

$$\ell_1:\ y = \tfrac{1}{3}x$$

💡 Turning the slope (rise $1$, run $3$) plus the $y$-intercept $0$ into $y = mx + b$ is the Grade 8 linear-function move.

#7 Identify Subproblems 8.F.A.3 Step 3

Check $\ell_1$ against the rectangle as a separate subproblem (Tool #7).
At the left edge $x=15$, line $\ell_1$ is at $y = \tfrac{15}{3} = 5$.
At the right edge $x=16$, it is at $y = \tfrac{16}{3} \approx 5.33$.
So as we sweep across the rectangle's $x$-strip the line's height runs from $5$ up to about $5.33$, completely above the rectangle's $y$-band $[3,5]$ except for one shared height: $y=5$, hit only at $x=15$.
That single shared point is exactly the corner $(15, 5)$.

$$\ell_1(15)=5,\quad \ell_1(16)=\tfrac{16}{3}\approx 5.33\ \Rightarrow\ \text{touches at } (15,5)\text{ only}$$

💡 Evaluating $y=mx+b$ at two specific $x$ values is the Grade 8 linear-function evaluation, then we just compare ranges.

#7 Identify Subproblems 8.F.A.3 Step 4

Now do the same subproblem for $\ell_2$ through $C(0,10)$ and $D(2,9)$.
From $C$ to $D$ the line drops $1$ for every $2$ steps right, so the slope is $-\tfrac{1}{2}$ and the $y$-intercept is $10$, giving $y = -\tfrac{1}{2}x + 10$.
At $x=15$: $y = -7.5 + 10 = 2.5$.
At $x=16$: $y = -8 + 10 = 2$.
So inside the rectangle's $x$-strip the line's height is between $2$ and $2.5$ — entirely below the band $[3,5]$.
No intersection at all.

$$\ell_2:\ y=-\tfrac{1}{2}x+10;\ \ell_2(15)=2.5,\ \ell_2(16)=2$$

💡 Again it is just "plug in $x$ to $y=mx+b$ and compare," the Grade 8 linear-function skill.

#3 Eliminate Possibilities 5.G.A.2 Step 5

Combine the two subproblems and pick the answer (Tool #3).
Line $\ell_1$ contributes $1$ point — the corner $(15,5)$.
Line $\ell_2$ contributes $0$ points.
The single point from $\ell_1$ is not shared with $\ell_2$ (since $\ell_2$ misses the rectangle entirely), so the total count is $1 + 0 = 1$.
Among (A) 0, (B) 1, (C) 2, (D) 3, (E) 4, only (B) matches.

$$1 + 0 = 1 \;\Rightarrow\; \textbf{(B)}$$

💡 Comparing the two single points we counted is the Grade 5 "reading a coordinate-plane situation" skill.

[1] #1 6.G.A.3 Read the rectangle as a window. Because the sides are parallel to the axes and t

[2] #1 8.F.A.3 Find the rule for line $\ell_1$ through $A(0,0)$ and $B(3,1)$. Going from $A$ to

[3] #7 8.F.A.3 Check $\ell_1$ against the rectangle as a separate subproblem (Tool #7). At the

[4] #7 8.F.A.3 Now do the same subproblem for $\ell_2$ through $C(0,10)$ and $D(2,9)$. From $C$

[5] #3 5.G.A.2 Combine the two subproblems and pick the answer (Tool #3). Line $\ell_1$ contrib

Review

Reasonableness: Does $1$ make sense? Line $\ell_1$ has slope $\tfrac{1}{3}$ starting from the origin, so at $x=15$ it should be at $y=5$ exactly — and the corner $(15,5)$ of the rectangle sits at height $5$, so a glancing touch at one corner is exactly what we expect. Line $\ell_2$ starts high at $(0,10)$ and drops by $\tfrac{1}{2}$ per unit, so by $x=15$ it has fallen below $3$ — it shoots under the rectangle entirely. One point total is the consistent answer.

Alternative: We could lean on Tool #3 (Eliminate Possibilities) from the very start: choices (D) $3$ and (E) $4$ would require a line to lie along an entire side or corner-to-corner across the rectangle, which neither line's slope ($\tfrac{1}{3}$ or $-\tfrac{1}{2}$) allows — a straight line that is not vertical or horizontal can cross a convex rectangle in at most $2$ boundary points, and only $\ell_1$ even reaches the rectangle's $y$-band. That narrows it to (A), (B), or (C), and the corner-touch check picks (B).

CCSS standards used (min grade 8)

6.G.A.3 Draw polygons in the coordinate plane given coordinates for the vertices (Reading the rectangle as the region $15 \le x \le 16,\ 3 \le y \le 5$ from its two opposite vertices.)
5.G.A.2 Represent real-world and mathematical problems by graphing points (Plotting and interpreting the corner $(15,5)$ as a single shared point between the line and the rectangle.)
8.F.A.3 Interpret the equation y = mx + b as defining a linear function (Writing $\ell_1: y = \tfrac{1}{3}x$ and $\ell_2: y = -\tfrac{1}{2}x + 10$ from two points each, then evaluating at $x = 15$ and $x = 16$ to see if the line enters the rectangle's $y$-band.)

⭐ This AMC 8 problem only needs Grade 8 linear functions (the $y = mx + b$ rule) you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 8)

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