Problem
Equal point distances make isosceles triangles on a peg board
On a 3-by-3 square peg board (9 evenly spaced dots), I must count every isosceles triangle whose three corners are dots on the board.
Geometry
Your answer
Watch out It's easy to find the right isosceles triangles with legs of length 1 or 2, but miss the tilted ones whose two equal sides are slanted (diagonal) segments of the same length.
How to solve
Strategy Make a Systematic List — This is a 'how many' counting question over a small finite board, so I list cases systematically. Drawing the board lets me measure distances by counting grid steps, and grouping triangles by their apex (the dot where the two equal sides meet) keeps the list organized and avoids double counting.
1STEP 1
Set up the grid and how to compare side lengths
Plot the dots at (0,0)-(2,2); segments with the same grid-step pattern are equal in length.
dots at (x,y), x,y ∈ {0,1,2}
Grade 4 students can plot points and compare segment lengths by counting equal grid steps.
4.G.A.1Draw A DiagramI can decide whether two sides of a triangle are equal in length just by checking that they cover the same across-and-up grid steps, instead of measuring with a ruler.
Why?
Two board segments built from the same across amount and up amount are really one segment set down in a new place, so they must be equal in length.
Why?
When both segments go the same way, like 1 across and 2 up, one can be slid straight over until it sits exactly on the other.
Why?
Sliding a segment moves it as one stiff piece and drops it exactly onto the other, and two things laid exactly on each other have the same length.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
When one segment goes 1 across and 2 up while the other goes 2 across and 1 up, they are mirror images, so flipping one over lays it exactly on the other.
Why?
Flipping a segment over is just a fold, and a fold lays one part exactly onto the matching part, keeping its length the same.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
2STEP 2
Count triangles whose apex sits at a corner dot
Each of the 4 corner dots is the apex of 3 isosceles triangles (including tilted ones): 4 × 3 = 12.
Sorting by apex dot is a tidy way to list cases so none are missed or counted twice.
4.G.A.2Make A Systematic List3STEP 3
Count triangles whose apex sits at an edge-midpoint dot or the center dot
Each edge-middle dot is the apex of 4, and the center dot of 8: 4 × 4 + 8 = 24 more.
The same equal-distance idea, just applied to every dot in turn, keeps the count complete.
4.G.A.2Make A Systematic List4STEP 4
Add up all the isosceles triangles
No triangle repeats across apexes (no equilateral on a square grid), so 12 + 24 = 36.
isosceles total = 36
Knowing the grid has no equilateral triangles means no triangle is double-counted, so the apex groups simply add up.
4.G.A.2Count The ComplementAnswer
36 isosceles triangles
There are 76 triangles in total on a 3x3 board (84 ways to pick 3 dots minus 8 straight-line triples). Getting 36 isosceles, a bit under half, is sensible: many grid triangles are scalene, but the symmetric board produces lots of equal-distance pairs, so a large minority being isosceles fits.
Takeaway
This only needs Grade 4 point-plotting and careful, organized counting you already know!
- Set up the grid and how to compare side lengths
- Count triangles whose apex sits at a corner dot
- Count triangles whose apex sits at an edge-midpoint dot or the center dot
- Add up all the isosceles triangles
Where next?
Level upsuggested
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested