Reasoning · Grade 5-2 Using Symmetry

Problem

Shortest path that must touch a line

A horizontal line carries points A and B, with A on the left. Taking C as the origin, with right and up positive, A is at (8, 15.5), B at (21.5, 15.5) and D at (37, 4.5), so C and D both lie below the line. The red path runs C to A to D, the blue path C to B to D. Decide which path is shorter.
A B C D
Your answer
How to solve
Strategy Draw a Diagram — Bent paths are hard to compare because you cannot lay a ruler along them in one go. The fix is to redraw the picture so that the bend disappears. Fold the paper along the line and prick through the point D: it lands at a new point D' the same distance above the line as D was below it. Because folding does not change lengths, the second half of each path, AD or BD, has exactly the same length as AD' or BD'. Each bent path from C to D therefore has the same length as a path from C to D', and now the question is the easy one every ruler can answer: which path from C to D' is closer to being the straight segment CD'? Finally I check the verdict by measuring both paths off the drawing.
1STEP 1

Read the four points off the drawing

Take the four points' positions from the coordinates given.

C=(0,0), A=(8, 15.5), B=(21.5, 15.5), D=(37, 4.5)
2STEP 2

Fold along the line to find the mirror point D'

Fold to mark D's mirror point.

D' = (37, 26.5), AD = AD', BD = BD'
3STEP 3

Straighten both paths

Lengths survive the fold, so both paths straighten out.

CA + AD = CA + AD', CB + BD = CB + BD'
4STEP 4

See which turning point lies on the straight segment

The turning point closer to the straight segment is B.

x = (37 × 15.5)/26.5 ≈ 21.6 versus x_B = 21.5, x_A = 8
5STEP 5

Measure the red path

The red path measures about 48.5.

CA = √(8²+15.5²) = √(304.25) ≈ 17.44, AD = √(29²+11²) = √(962) ≈ 31.02, red ≈ 48.46
6STEP 6

Measure the blue path and compare

The blue path is about 45.5, so it is shorter.

CB = √(21.5²+15.5²) = √(702.5) ≈ 26.50, BD = √(15.5²+11²) = √(361.25) ≈ 19.01, blue ≈ 45.51 ＜ 48.46
Answer
the blue path
Every length is a distance in the same grid units used to read the drawing, and each one is longer than either of its two legs but shorter than their sum, exactly as a slant side must be — for instance CB ≈ 26.5 sits between 21.5 and 21.5+15.5=37. As a lower bound, no path from C to D that has to touch the line can be shorter than the straight segment CD', which measures √(37²+26.5²) = √(2071.25) ≈ 45.511. The blue path comes out at about 45.5113 — just a thousandth above that floor, since B sits almost exactly where the straight segment crosses. Nothing came out below the floor, so the numbers hold together. Both answers are also comfortably longer than the direct distance CD ≈ 37.3, which is right, because the detour up to the line has to cost something.
Takeaway

Fold the paper along the line so the far point flips to the near side — the bent path becomes a straight one, and the path that turns closest to that straight line wins.

  • Read the four points off the drawing
  • Fold along the line to find the mirror point D'
  • Straighten both paths
  • See which turning point lies on the straight segment
  • Measure the red path
  • Measure the blue path and compare