AMC 10 · 2009 · #12
Grade 6 geometry-2dDistinct points A, B, C, and D lie on a line, with AB=BC=CD=1. Points E and F lie on a second line, parallel to the first, with EF=1. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?
Pick an answer.
AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Four distinct points $A,B,C,D$ sit on one line with $AB=BC=CD=1$. Two more points $E,F$ sit on a second line parallel to the first, with $EF=1$. Pick any $3$ of these $6$ points that form a triangle with positive area. Count how many different values the area of that triangle can take.
Givens: $A,B,C,D$ are on one line, evenly spaced one unit apart, so they can be placed at positions $0,1,2,3$.; $E,F$ are on a second line parallel to the first, with $EF=1$.; The two lines are a fixed distance apart; call that distance $h$.; A triangle is formed by choosing $3$ of the $6$ points and must have positive area.; Answer choices: (A) $3$, (B) $4$, (C) $5$, (D) $6$, (E) $7$.
Unknowns: The number of distinct possible values for the triangle's area.
Understand
Restated: Four distinct points $A,B,C,D$ sit on one line with $AB=BC=CD=1$. Two more points $E,F$ sit on a second line parallel to the first, with $EF=1$. Pick any $3$ of these $6$ points that form a triangle with positive area. Count how many different values the area of that triangle can take.
Givens: $A,B,C,D$ are on one line, evenly spaced one unit apart, so they can be placed at positions $0,1,2,3$.; $E,F$ are on a second line parallel to the first, with $EF=1$.; The two lines are a fixed distance apart; call that distance $h$.; A triangle is formed by choosing $3$ of the $6$ points and must have positive area.; Answer choices: (A) $3$, (B) $4$, (C) $5$, (D) $6$, (E) $7$.
Plan
Primary tool: #1 Draw a Diagram
Secondary: #7 Identify Subproblems, #2 Make a Systematic List
A picture of the two parallel lines (Tool #1) reveals the hidden structure: every triangle has one whole side sitting on a parallel line, and its third vertex on the other line. Because parallel lines stay the same distance apart, the height of every such triangle is the same. That collapses the whole problem into one question about the base. Tool #7 (Identify Subproblems) splits the base into two cases (base on the crowded line vs. base on the two-point line), and Tool #2 (Make a Systematic List) enumerates the distinct base lengths, which map one-to-one onto the distinct areas.
Execute — Answer: A
4.G.A.1 Step 1 Draw the two parallel lines
- Put $A,B,C,D$ on the bottom line at positions $0,1,2,3$ (they are one unit apart).
- Put $E,F$ on the top line, one unit apart, and let the gap between the two lines be $h$.
- A triangle needs $3$ points that are not all on one line.
- Since $4$ points crowd the bottom line and only $2$ sit on the top line, any triangle must borrow points from both lines: either $2$ from the bottom and $1$ from the top, or $1$ from the bottom and $2$ from the top.
💡 Drawing the two parallel lines shows at once that no triangle can be built from a single line, so every triangle straddles both.
6.G.A.1 Step 2 The height is always the same
- Choose the side of the triangle that lies on one of the parallel lines as the base.
- The remaining vertex sits on the other parallel line.
- The distance from that vertex to the base line is just the gap between the two parallel lines, which is $h$ no matter which points you picked.
- So every triangle here has the same height $h$, and its area is $\tfrac{1}{2}\,(\text{base})\,h$.
- Only the base length can change the area.
💡 Parallel lines never get closer or farther apart, so the far vertex is always the same height above the base.
6.NS.C.7 Step 3 Split the base into two cases
- There are only two places the base can lie.
- Case 1: the base is on the bottom line, joining two of the points at $0,1,2,3$.
- Its length is the distance between those positions, which is the absolute difference and can be $1$, $2$, or $3$.
- Case 2: the base is on the top line, but the only two points there are $E$ and $F$, so the base is $EF=1$.
💡 Breaking the base into 'which line is it on' captures every triangle without missing or repeating any.
6.G.A.1 Step 4 List the distinct areas and count
- Gather all possible base lengths from both cases: $\{1,2,3\}$ from the bottom line and $\{1\}$ from the top line.
- Together the distinct base lengths are $1$, $2$, and $3$.
- Since the area is $\tfrac{1}{2}\cdot \text{base}\cdot h$ with $h$ fixed, each distinct base gives a distinct area: $\tfrac{h}{2},\ h,\ \tfrac{3h}{2}$.
- That is $3$ possible values, so the answer is (A).
💡 With the height locked, counting areas is the same as counting how many different base lengths exist.
4.G.A.1 Put $A,B,C,D$ on the bottom line at positions $0,1,2,3$ (they are one unit apart 6.G.A.1 Choose the side of the triangle that lies on one of the parallel lines as the ba 6.NS.C.7 There are only two places the base can lie. Case 1: the base is on the bottom li 6.G.A.1 Gather all possible base lengths from both cases: $\{1,2,3\}$ from the bottom li Review
Reasonableness: The base on the top line ($EF=1$) matches a base already available on the bottom line (the unit gap like $AB$), so it adds no new area — a good sign the count stays small. The only base lengths anywhere are $1$, $2$, and $3$, giving exactly $3$ areas $\tfrac{h}{2}, h, \tfrac{3h}{2}$. Choice (A) $3$ fits; the larger choices would require base lengths that simply do not exist among these points.
Alternative: Instead of reasoning about height, one could list a few actual triangles and compute areas with $h$ left as a symbol: e.g. $A,B,E$ gives base $1$, area $\tfrac{h}{2}$; $A,C,E$ gives base $2$, area $h$; $A,D,E$ gives base $3$, area $\tfrac{3h}{2}$; $A,E,F$ gives base $1$, area $\tfrac{h}{2}$ (a repeat). Every triangle reduces to one of these three areas, confirming $3$.
CCSS standards used (min grade 6)
4.G.A.1Draw points, lines, line segments, rays, angles, and identify in figures (Setting up the diagram of two parallel lines and placing the six points to see that every triangle spans both lines.)6.G.A.1Find area of triangles, special quadrilaterals, and polygons by composing (Using Area $=\tfrac{1}{2}\cdot\text{base}\cdot\text{height}$ with a fixed height to tie each distinct area to a distinct base length.)6.NS.C.7Understand ordering and absolute value of rational numbers (Finding the base lengths on the bottom line as absolute differences of the positions $0,1,2,3$, giving $1,2,3$.)
⭐ When all your triangles have a side on one of two parallel lines, the height never changes, so counting different areas is just counting different base lengths.
⭐ When all your triangles have a side on one of two parallel lines, the height never changes, so counting different areas is just counting different base lengths.
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