Reasoning · Grade 5-2 Congruent Figures
Problem
The moves are joining midpoints and joining the centre.
Naming the two moves first turns six separate drawing puzzles into a choice between two familiar cuts, which is much less to keep in your head.
3.G.A.2Draw A DiagramTwo cutting moves generate every case: halving along a median, and splitting into four by joining the midpoints.
Both moves cut the triangle into pieces that are all the same size, which is exactly what equal shares means.
The pieces fill the triangle with no gap and no overlap, so applying a move to every piece keeps the whole covered.
Apex to base midpoint gives 2 pieces.
The line down from the apex is the triangle's own mirror line, so the two halves must match - and matching after a flip still counts as congruent.
3.G.A.2Draw A DiagramCentre to three vertices gives 3 pieces.
Angles round a point add to a full turn, so three equal angles at the centre must be 120 degrees each - which is exactly the turn that carries one piece onto the next.
4.MD.C.7Draw A DiagramJoining the three midpoints gives 4 pieces.
Halving every side halves every small triangle's side, and equilateral triangles with equal sides are automatically the same shape and size.
3.G.A.2Look For A PatternThirds along each side give 9 pieces.
Once you see 4 = 2 x 2 for halves, the same picture with thirds gives 3 x 3 = 9, and the upside-down pieces are just half-turns of the upright ones.
3.OA.A.3Look For A PatternHalving each of the three gives 6 pieces.
Halving each of three matching pieces in the same way keeps the halves matching, so doubling a count you already have is easier than inventing a brand new cut.
3.OA.A.3Solve An Easier Related ProblemHalving each of the four gives 8 pieces.
Every one of the eight pieces is made the same way out of an identical small triangle, so they all come out identical without any new measuring.
3.OA.A.3Solve An Easier Related ProblemAll six cuts give congruent pieces.
Congruent pieces must have equal areas, so counting the pieces and checking they exactly fill the triangle is a quick way to catch a cut that has gone wrong.
3.G.A.2Draw A DiagramFind one cut that works, then halve it again - 2 becomes 4 becomes 8, and 3 becomes 6, without inventing anything new!