AMC 10 · 2011 · #9
Grade 7 geometry-2dA rectangular region is bounded by the graphs of the equations y=a,y=−b,x=−c, and x=d, where a,b,c, and d are all positive numbers. Which of the following represents the area of this region?
Pick an answer.
AMC 10 2011 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: In the coordinate plane, four lines form a rectangle: the horizontal lines $y=a$ (on top) and $y=-b$ (on the bottom), and the vertical lines $x=-c$ (on the left) and $x=d$ (on the right). All of $a$, $b$, $c$, $d$ are positive. Find an expression for the area of the rectangle.
Givens: Top edge is the line $y=a$; Bottom edge is the line $y=-b$; Left edge is the line $x=-c$; Right edge is the line $x=d$; $a$, $b$, $c$, and $d$ are all positive numbers
Unknowns: An algebraic expression for the area of the rectangle
Understand
Restated: In the coordinate plane, four lines form a rectangle: the horizontal lines $y=a$ (on top) and $y=-b$ (on the bottom), and the vertical lines $x=-c$ (on the left) and $x=d$ (on the right). All of $a$, $b$, $c$, $d$ are positive. Find an expression for the area of the rectangle.
Givens: Top edge is the line $y=a$; Bottom edge is the line $y=-b$; Left edge is the line $x=-c$; Right edge is the line $x=d$; $a$, $b$, $c$, and $d$ are all positive numbers
Plan
Primary tool: #1 Draw a Diagram
Secondary: #13 Convert to Algebra, #7 Identify Subproblems
The four equations are just lines, so I sketch them to see the rectangle they trap. Once the picture is clear, the width and height are horizontal and vertical distances I can read straight off the axes, and the area is their product. I then turn that product into one of the listed expressions by expanding it.
Execute — Answer: A
6.NS.C.6 Step 1 Sketch the four boundary lines
- Draw $y=a$ as a horizontal line above the $x$-axis and $y=-b$ as a horizontal line below it.
- Draw $x=d$ as a vertical line right of the $y$-axis and $x=-c$ as a vertical line left of it.
- These four lines cross to make a rectangle straddling the origin.
💡 Equations like $y=a$ or $x=d$ each pin down one straight line, and a rectangle is just four such lines meeting at corners.
6.NS.C.8 Step 2 Find the width
- The width is the horizontal gap between the left edge $x=-c$ and the right edge $x=d$.
- The two lines sit on opposite sides of the $y$-axis, so their distances add: the width is $d-(-c)=c+d$.
💡 For two vertical lines the distance is the difference of their $x$-values, and subtracting a negative adds the two positive distances from the axis.
6.NS.C.8 Step 3 Find the height
- The height is the vertical gap between the bottom edge $y=-b$ and the top edge $y=a$.
- They lie on opposite sides of the $x$-axis, so again the distances add: the height is $a-(-b)=a+b$.
💡 The same distance rule works vertically: the difference of the two $y$-values, and again subtracting a negative adds the pieces.
3.MD.C.7 Step 4 Write the area as a product
- The area of a rectangle is width times height.
- Using the width and height I found, the area is $(c+d)(a+b)$, which I will write as $(a+b)(c+d)$.
💡 Length times width gives the area of any rectangle, even when the sides are written as sums of letters.
7.EE.A.1 Step 5 Expand to match a choice
- Multiply the two sums by distributing every term of $(a+b)$ across every term of $(c+d)$: $a\cdot c + a\cdot d + b\cdot c + b\cdot d$.
- That gives $ac+ad+bc+bd$, which is choice (A).
💡 Each part of the first sum has to multiply each part of the second, producing all four products.
6.NS.C.6 Draw $y=a$ as a horizontal line above the $x$-axis and $y=-b$ as a horizontal li 6.NS.C.8 The width is the horizontal gap between the left edge $x=-c$ and the right edge 6.NS.C.8 The height is the vertical gap between the bottom edge $y=-b$ and the top edge $ 3.MD.C.7 The area of a rectangle is width times height. Using the width and height I foun 7.EE.A.1 Multiply the two sums by distributing every term of $(a+b)$ across every term of Review
Reasonableness: Every term $ac$, $ad$, $bc$, $bd$ is a product of two positive numbers, so each is positive and the whole area is positive — exactly what an area must be. Choices with minus signs (B, C, D, E) could turn negative for some positive $a,b,c,d$, so they cannot be areas. Only (A) is a sum of all four positive products, confirming the answer.
Alternative: Plug in easy numbers. Let $a=b=c=d=1$; then the rectangle runs from $x=-1$ to $x=1$ (width 2) and $y=-1$ to $y=1$ (height 2), so its area is $4$. Testing the choices, only (A) gives $1+1+1+1=4$; the others give $0$, so (A) is the answer.
CCSS standards used (min grade 7)
6.NS.C.6Understand a rational number as a point on the number line (Placing the lines $y=a$, $y=-b$, $x=-c$, $x=d$ correctly relative to the axes using the signs of the numbers.)6.NS.C.8Solve real-world problems by graphing points in all four quadrants (Finding the width $c+d$ and height $a+b$ as distances between lines on opposite sides of each axis.)3.MD.C.7Relate area to multiplication and addition operations (Writing the rectangle's area as width times height, $(a+b)(c+d)$.)7.EE.A.1Apply properties of operations to add, subtract, factor, and expand linear expressions (Expanding $(a+b)(c+d)$ into $ac+ad+bc+bd$ to match a listed choice.)
⭐ Draw the four lines, read the width and height off the axes as $c+d$ and $a+b$, then multiply and expand to get $ac+ad+bc+bd$.
⭐ Draw the four lines, read the width and height off the axes as $c+d$ and $a+b$, then multiply and expand to get $ac+ad+bc+bd$.
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