Distance between parallel sides equals the side length

4.G.A.14.MD.A.3 · adapt · grade 4

Archetype: Perimeter by Tracing Every Side · step in a 11-type progression

Three squares of different sizes — A, B, and C — are joined side by side without overlapping. Side $\overline{\text{GH}}$ (the left side of square A) is parallel to side $\overline{\text{KJ}}$ (the right side of square C). Find the distance, in $\text{cm}$ , between side $\overline{\text{GH}}$ and side $\overline{\text{KJ}}$ .

[Figure] Three squares of different sizes — A, B, and C — sit in a row with their bottom edges aligned, joined left to right in the order A, B, C. The largest square, A, has its left side labeled G (top) and H (bottom), with length $9\,\text{cm}$ . The difference in height between the top edge of square A and the top edge of square B is marked as $2\,\text{cm}$ , and the difference in height between the top edge of square B and the top edge of square C is also marked as $2\,\text{cm}$ . The rightmost square, C, has its right side labeled K (top) and J (bottom).

Show solution

Understand

Three squares A, B, C are lined up left to right with their bottoms aligned. The left side of A (segment GH) is 9 cm tall. A's top is 2 cm higher than B's top, and B's top is 2 cm higher than C's top. I need the horizontal distance between segment GH (left side of A) and segment KJ (right side of C).

Givens

Squares A, B, C joined side by side, bottoms aligned
Left side GH of square A is 9 cm
A's top edge is 2 cm higher than B's top edge
B's top edge is 2 cm higher than C's top edge
GH (left side of A) is parallel to KJ (right side of C)

Unknowns

The distance in cm between side GH and side KJ

Constraints

Each shape is a square, so all four of its sides are equal
The squares do not overlap and sit in a single row

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Since the bottoms are aligned, the height difference between two adjacent square tops equals the difference of their side lengths. I first find each square's side as a small subproblem, then add the three sides because GH and KJ are the two outer vertical sides, so the gap between them is just the total width of the row.

Execute

#7 Identify Subproblems 4.G.A.1

Square A's left side GH is given as 9 cm, and a square has all sides equal, so A's side length is 9 cm.

a = 9\text{ cm}

Knowing a square has four equal sides is basic shape sense.

#7 Identify Subproblems 4.MD.A.3

Bottoms are aligned, so the 2 cm gap between A's top and B's top is exactly how much shorter B's side is than A's side. So B's side is 9 - 2 = 7 cm.

b = 9 - 2 = 7\text{ cm}

When two squares share a baseline, the height step between their tops is the difference of their sides.

#7 Identify Subproblems 4.MD.A.3

B's top is 2 cm higher than C's top, so C's side is 2 cm shorter than B's: 7 - 2 = 5 cm.

c = 7 - 2 = 5\text{ cm}

Same step idea applied to the next neighboring pair.

#1 Draw a Diagram 4.MD.A.3

GH is the far-left vertical side and KJ is the far-right vertical side. The distance between these two parallel sides is the whole row's width, which is the three square sides placed end to end: 9 + 7 + 5.

9 + 7 + 5 = 21\text{ cm}

A picture shows the left and right sides bound the full strip, so the gap equals the summed widths.

Answer: 21 cm

Review

The three sides are 9, 7, and 5 cm, each a sensible square size, and 9 + 7 + 5 = 21 cm. The answer is larger than any single square width, which is right because GH and KJ sit at opposite ends of all three squares.

Guess and check (tool 6): try a side for B, verify the 2 cm steps, and adjust until the top-edge gaps both equal 2 cm; this leads to the same 9, 7, 5 sides and total 21 cm.

Standards · min grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Recognizing the square's equal sides and the named segments GH and KJ
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using side lengths and the aligned baseline to find each square's width and sum them

💡 Lined-up squares: the step between their tops is the difference of their sides, so you just add the widths!