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← 4-2 · 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

Generated variants — 3

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

Variant 1 answer: 36 isosceles triangles

The figure below shows 99 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 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.

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.
dots at (x,y), x,y∈{0,…,2}\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.
#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.
#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.
isosceles total=36\text{isosceles total} = 36
Counting the complement (all triangles) frames how many of them turn out to be isosceles.
Answer: 36 isosceles triangles

Review

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.

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.

Standards · min 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.
💡 This only needs Grade 4 point-plotting and careful, organized counting you already know!
Variant 2 answer: 4 isosceles triangles

The figure below shows 44 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 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)

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 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.

Execute

#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.
dots at (x,y), x,y∈{0,…,1}\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.
#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.
#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.
isosceles total=4\text{isosceles total} = 4
Counting the complement (all triangles) frames how many of them turn out to be isosceles.
Answer: 4 isosceles triangles

Review

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.

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.

Standards · min 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.
💡 This only needs Grade 4 point-plotting and careful, organized counting you already know!
Variant 3 answer: 148 isosceles triangles

The figure below shows 1616 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 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)

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 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.

Execute

#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.
dots at (x,y), x,y∈{0,…,3}\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.
#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.
#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.
isosceles total=148\text{isosceles total} = 148
Counting the complement (all triangles) frames how many of them turn out to be isosceles.
Answer: 148 isosceles triangles

Review

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.

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.

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