← Equal point distances make isosceles triangles on a peg board · Isosceles and Equilateral Angle Chaining

Equal point distances make isosceles triangles on a peg board · 3 practice problems

4.G.A.14.G.A.2

0 / 3 solved

Show all answers (parent / teacher)

Generated variants — 3

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

The figure below shows $4$ dots arranged in a square pattern. Using these dots as vertices, how many isosceles triangles can be made in all?

Your answer

Show solution

1 · Understandwhat's really being asked

On a 2-by-2 square peg board (4 evenly spaced dots), I must count every isosceles triangle whose three corners are dots on the board.

Givens

4 dots arranged in 2 rows and 2 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)

2 · Planchoose the strategy

#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 compare 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.

3 · Execute3 carry out the plan

1Set up the grid and how to compare side lengths

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

Place the dots at coordinates (0,0) up to (1,1). 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,\dots,1\}

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

2List triangles by their apex dot

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

For each dot in turn, treat it as the apex and look for two board segments of equal length meeting there that also close into a triangle. Going dot by dot keeps the list complete and avoids missing or repeating cases.

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

3Add up all the isosceles triangles

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

Combining every apex group, the systematic list totals 4 isosceles triangles. As a sanity frame, the board makes 4 triangles in all, and exactly 4 of them are isosceles.

\text{isosceles total} = 4

Counting the complement (all triangles) frames how many of them turn out to be isosceles.

Answer: 4 isosceles triangles

4 · Reviewdoes it hold up?

There are 4 triangles in total on a 2x2 board. Getting 4 isosceles is sensible: the symmetric board produces many equal-distance pairs, so a large share being isosceles fits.

Another way: Instead of listing by apex, solve an easier related problem first (a smaller board), notice the pattern of equal-distance pairs, then scale the reasoning up.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Plotting the 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.

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

Variant 2 medium

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?

Your answer

Show solution

1 · Understandwhat's really being asked

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)

2 · Planchoose the strategy

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

3 · Execute3 carry out the plan

1Set up the grid and how to compare side lengths

#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,\dots,2\}

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

2List triangles by their apex dot

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

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

3Add up all the isosceles triangles

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

Combining every apex group, 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

Counting the complement (all triangles) frames how many of them turn out to be isosceles.

Answer: 36 isosceles triangles

4 · Reviewdoes it hold up?

There are 76 triangles in total on a 3x3 board. Getting 36 isosceles is sensible: the symmetric board produces many equal-distance pairs, so a large share being isosceles fits.

Another way: Instead of listing by apex, solve an easier related problem first (a smaller board), notice the pattern of equal-distance pairs, then scale the reasoning up.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Plotting the 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.

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

Variant 3 hard

The figure below shows $16$ dots arranged in a square pattern. Using these dots as vertices, how many isosceles triangles can be made in all?

Your answer

Show solution

1 · Understandwhat's really being asked

On a 4-by-4 square peg board (16 evenly spaced dots), I must count every isosceles triangle whose three corners are dots on the board.

Givens

16 dots arranged in 4 rows and 4 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)

2 · Planchoose the strategy

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

3 · Execute3 carry out the plan

1Set up the grid and how to compare side lengths

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

Place the dots at coordinates (0,0) up to (3,3). 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,\dots,3\}

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

2List triangles by their apex dot

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

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

3Add up all the isosceles triangles

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

Combining every apex group, the systematic list totals 148 isosceles triangles. As a sanity frame, the board makes 516 triangles in all, and exactly 148 of them are isosceles.

\text{isosceles total} = 148

Counting the complement (all triangles) frames how many of them turn out to be isosceles.

Answer: 148 isosceles triangles

4 · Reviewdoes it hold up?

There are 516 triangles in total on a 4x4 board. Getting 148 isosceles is sensible: the symmetric board produces many equal-distance pairs, so a large share being isosceles fits.

Another way: Instead of listing by apex, solve an easier related problem first (a smaller board), notice the pattern of equal-distance pairs, then scale the reasoning up.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Plotting the 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.

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

Where next?

← Back to the problem: Equal point distances make isosceles triangles on a peg board