Numbers & Place Value

Depth · one concept from first principles · step 2 of 4

Three reflections make a rectangle sized by absolute values

Reflect a lattice point P(a, b) through the origin and across both axes. The four points form a quadrilateral; count the integer pairs (a, b) that give it perimeter 12.
Number system
Your answer
How to solve
Strategy Draw a Diagram — Draw the four points: they are the corners of a rectangle centered at the origin with sides parallel to the axes. Its side lengths depend only on |a| and |b|, so the perimeter condition becomes one short equation in |a| and |b|. Then list its solutions and attach signs.
1STEP 1

Write the reflected points

Reflecting through the origin flips both signs, across the x-axis flips the y-sign, and across the y-axis flips the x-sign.

A(-a, -b), B(a, -b), C(-a, b)
2STEP 2

See the rectangle

The points (± a, ± b) form a rectangle 2|a| wide and 2|b| tall; it is a real quadrilateral only if a ≠ 0 and b ≠ 0.

|a-(-a)| = 2|a|, |b-(-b)| = 2|b|
3STEP 3

Turn the perimeter into an equation

Two widths and two heights add to 12, so 4(|a| + |b|) = 12.

2(2|a| + 2|b|) = 12 → |a| + |b| = 3
4STEP 4

List the absolute values

With |a| and |b| both positive integers adding to 3, the only choices are 1+2 and 2+1.

(|a|, |b|) = (1, 2) or (2, 1)
5STEP 5

Attach the signs

Each absolute-value pair gives 2 signs for a and 2 for b, so 4 choices; 2 pairs give 8 ordered pairs.

2 × 4 = 8
Answer
8
The eight pairs are (± 1, ± 2) and (± 2, ± 1). For example (1, 2) gives a 2 × 4 rectangle with perimeter 2 × (2+4) = 12.
Takeaway

Reflections only flip signs -- so the size of the shape lives in the absolute values.

  • Write the reflected points
  • See the rectangle
  • Turn the perimeter into an equation
  • List the absolute values
  • Attach the signs
Where next?
Level upsuggested
Read each runner's rate from one graph point
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
Opposite signs turn |b|=2|a| into b=-2aCombine sign cases to get extreme distances

A parent dashboard for the family lives at sensimlab.com.

▶ Practice — 12 problems