AMC 10 · 2011 · #9

Grade 7 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Axis-Aligned Figure Area from Coordinates
coordinate-geometryarea-rectanglesabsolute-value convert-to-algebra ↑ Prerequisites: coordinate-geometry
📏 Medium solution 💡 2 insights
Problem
In the coordinate plane, a rectangular region is bounded by 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, where a, b, c, and d are all positive numbers. Which expression represents the area of this region?

Pick an answer.

(A)
ac+ad+bc+bd
(B)
ac-ad+bc-bd
(C)
ac+ad-bc-bd
(D)
-ac-ad+bc+bd
(E)
ac-ad-bc+bd

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

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.

1STEP 1

Sketch the four boundary lines

Draw y=a above the x-axis, y=-b below it, x=d right of the y-axis, and x=-c left of it; the four lines box in a rectangle.

y=a, y=-b, x=-c, x=d
2STEP 2

Find the width

The width is the gap from x=-c to x=d. They straddle the y-axis, so the distances add: d-(-c)=c+d.

width=d-(-c)=c+d
3STEP 3

Find the height

The height is the gap from y=-b to y=a. They straddle the x-axis, so again the distances add: a-(-b)=a+b.

height=a-(-b)=a+b
4STEP 4

Write the area as a product

A rectangle's area is width times height, so the area is (c+d)(a+b), which I write as (a+b)(c+d).

Area=(c+d)(a+b)=(a+b)(c+d)
5STEP 5

Expand to match a choice

Distribute every term of (a+b) across (c+d): the four products give ac+ad+bc+bd, which is choice (A).

(a+b)(c+d)=ac+ad+bc+bd
Answer
ac+ad+bc+bd
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.
💡Key takeaway

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.

  • Sketch the four boundary lines
  • Find the width
  • Find the height
  • Write the area as a product
  • Expand to match a choice

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