Reasoning · Grade 3-1 Counting Figures Formed by Joining Dots
Problem
Counting right triangles on a dot grid
Twelve evenly spaced dots sit on a rectangle's four sides, with none at the corners. Each vertex must come from a different side. Triangles matching after a turn or flip count once. Find how many different right triangles are possible.
Your answer
Watch outCounting dot choices gives a far bigger number — the question wants how many non-congruent shapes.
How to solve
StrategyDraw a Diagram — I put the points on a grid (the rectangle is 5 wide and 3 tall in equal steps) so I can read coordinates and test right angles. Then I make a systematic list by which three sides I use: {top, bottom, left}, {top, bottom, right}, {top, left, right}, {bottom, left, right}. For each combination I check which point choices give a square corner, and at the end I merge triangles that are the same after a flip or turn into one kind.
1STEP 1
Set up coordinates
With empty corners the rectangle is 5 by 3.
top (1..4,3), bottom (1..4,0), left (0,1),(0,2), right (5,1),(5,2)
Drawing the dots on a grid turns 'is this corner square?' into a clear picture I can test point by point.
4.G.A.1Draw A Diagram
2STEP 2
List the side-combinations to check
Choosing three of the four sides gives 4 ways.
C(4, 3) = 4 side-trios
Grouping by which three sides are used keeps the search organized so no triangle is missed.
3.OA.A.1Make A Systematic List
3STEP 3
Find the right triangles and group congruent ones
Keep the right-angled ones and group congruents — many repeat.
(2,8,10), (2,18,20), (5,5,10), (10,10,20), (5,20,25) (squared side lengths; each: leg²+leg²=hyp²)
Using squared side lengths lets me confirm the right angle (the two legs squared add to the hypotenuse squared) and tell shapes apart without measuring messy slants.
4.G.A.2Identify Subproblems
4STEP 4
Count the distinct kinds
What survives is 5 distinct kinds.
distinct right triangles = 5
Since the problem says turns and flips count as the same, I report the number of truly different shapes, which is five.
2.G.A.1Make A Systematic List
Since turns and flips count as the same triangle, the answer is the number of genuinely different shapes, not of positions.
Why?
Turning or flipping a triangle keeps every side length and every angle, so the moved copy is the same shape in a new place.
🧱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?
Sorting the triangles into families that can be moved onto one another puts each triangle in exactly one family, so the families can be counted.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Answer
5 kinds
Each of the five shapes passes the right-angle test: 2+8=10, 2+18=20, 5+5=10, 10+10=20, 5+20=25 — in every case the two leg-squares add to the hypotenuse-square, so all five really are right triangles. Two of them are isosceles right triangles and three are not, so they are clearly different shapes; five distinct kinds is a sensible, small count for a grades 2-4 enumeration.
Takeaway
Put the dots on a grid, check each corner for a square angle, then merge the flips and turns — you find exactly five different right triangles!