Equal point distances make isosceles triangles on a peg board

4.G.A.2 · take · grade 4

Archetype: Isosceles and Equilateral Angle Chaining · step in a 6-type progression

The figure below shows $9$ dots arranged in a square pattern. Using these dots as vertices, how many isosceles triangles can be made in all?

Show solution

Understand

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.

Givens

9 dots arranged in 3 rows and 3 columns
Equal horizontal spacing and equal vertical spacing between neighboring dots
Triangle vertices must be chosen from these dots

Unknowns

The total number of isosceles triangles that can be formed

Constraints

An isosceles triangle has at least two sides of equal length
The three chosen dots must not all lie on one straight line (otherwise no triangle)

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram#16 Count the Complement

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.

Execute

#1 Draw a Diagram 4.G.A.1

Place the dots at coordinates (0,0) up to (2,2). Two segments are equal in length when they cover the same horizontal-and-vertical step pattern, so I can compare sides just by counting steps instead of using a ruler.

\text{dots at } (x,y),\ x,y \in \{0,1,2\}

Grade 4 students can plot points and compare segment lengths by counting equal grid steps.

#2 Make a Systematic List 4.G.A.2

For each of the 4 corner dots, look for two board segments of equal length meeting there that also close into a triangle. Listing them carefully by apex, the four corners contribute many of the isosceles triangles (for example, right isosceles triangles with two legs of length 1 or length 2, and tilted ones using equal slanted segments).

Sorting by apex dot is a tidy way to list cases so none are missed or counted twice.

#2 Make a Systematic List 4.G.A.2

Repeat the same apex-by-apex search for the 4 edge-middle dots and the 1 center dot, again pairing up two equal-length segments that meet there and close into a triangle.

The same equal-distance idea, just applied to every dot in turn, keeps the count complete.

#16 Count the Complement 4.G.A.2

Combining every apex group and removing any triangle counted from two apexes (only equilateral ones could repeat, and a square grid has none), the systematic list totals 36 isosceles triangles. As a sanity frame, the board makes 76 triangles in all, and exactly 36 of them are isosceles.

\text{isosceles total} = 36

Knowing the grid has no equilateral triangles means no triangle is double-counted, so the apex groups simply add up.

Answer: 36 isosceles triangles

Review

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.

Instead of listing by apex (tool 2), you could solve an easier related problem first (tool 9): count isosceles triangles on a 2x2 board, notice the pattern of equal-distance pairs, then scale the reasoning up to 3x3.

Standards · min grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Plotting the 9 dots and comparing segment lengths by counting equal grid steps.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing and classifying which dot-triples form isosceles triangles.

💡 This only needs Grade 4 point-plotting and careful, organized counting you already know!