Count the number of triangles and squares in the given figure.

Concept

To count shapes in composite figures built from overlapping lines, group them by size — how many elementary regions each one encloses — rather than counting one at a time. Count every congruent group of 1-part, 2-part, 4-part, … shapes separately using the figure's labelled points, then sum every group. This method finds every larger shape formed by combining smaller ones and avoids double-counting.

Applying it to this figure

Label the figure as shown: outer square ABCD with E, F, G, H as the midpoints of sides AB, BC, CD, DA; diagonals AC and BD meet at the centre O; joining the midpoints gives the inner square EFGH, and the outer diagonal segments AO, BO, CO, DO cross the inner square's sides HE, EF, FG, GH respectively at I, J, K, L.

Triangles

1. 1-part (simplest) triangles: AEI, EOI, OHI, HAI, EBJ, BFJ, FOJ, OEJ, HOL, OGL, GDL, DHL, OFK, FCK, CGK, GOK — 16 triangles.

2. 2-part triangles: HAE, AEO, EOH, OHA, OEB, EBF, BFO, FOE, DHO, HOG, OGD, GDH, GOF, OFC, FCG, CGO — 16 triangles.

3. 4-part triangles: HEF, EFG, FGH, GHE, ABO, BCO, CDO, DAO — 8 triangles.

4. 8-part triangles: DAB, ABC, BCD, CDA — 4 triangles.

Total triangles = 16 + 16 + 8 + 4 = 44.

Squares

1. 2-part squares: HIOL, IEJO, JFKO, KGLO — 4 squares.

2. 4-part squares: AEOH, EBFO, OFGC, HOGD — 4 squares.

3. 8-part square: EFGH — 1 square.

4. 16-part square: ABCD — 1 square.

Total squares = 4 + 4 + 1 + 1 = 10.

Cross-check

The figure has 4-fold rotational symmetry about O, so shape counts must come out as multiples of 4 in each size-group — and 16, 16, 8, 4 all do (the two biggest squares are each unique by construction). This confirms the totals below.

The figure contains 44 triangles and 10 squares.