Select the alternative which represents three out of the five alternative…

Concept

In a figure-assembly (dissection) question, the target shape is cut into a fixed set of pieces of specific shapes. A candidate fragment can be part of the answer only if it matches the RIGHT KIND of piece the dissection needs (the same vertex count and angle profile, e.g. triangular vs. quadrilateral) as well as the right SIZE (its cut edges equal in length to the edges it must close against), so the chosen pieces close up edge to edge with no gap and no overlap.

Application

The equilateral triangle shown is cut into three pieces: one triangular fragment carrying a right angle at the internal cut, and two quadrilateral fragments, each carrying one of the triangle's own corners plus a stretch of its outer sides.

• Shape (2) is the required triangular fragment: a right-angled triangle whose acute 60° vertex sits at one of the triangle's own corners.

• Shape (4) is one of the two required quadrilateral fragments, carrying the triangle's left-hand corner, with a straight edge equal in length to one of shape (2)'s cut edges.

• Shape (5) is the other required quadrilateral fragment, carrying the triangle's right-hand corner and part of the base, with a straight edge equal in length to shape (2)'s other cut edge.

Fitted together, shape (2)'s two cut edges close exactly against the matching edges of (4) and (5), and the three pieces' outer edges line up end to end to reproduce the triangle's three original sides — reconstructing the equilateral triangle shown in the assembled figure below, with no gap or overlap.

Cross-check

The dissection needs exactly one triangular fragment and two quadrilateral fragments, with the cut edges matching in length. Shape (1) is also a plain triangle like (2), but it has no right angle, so its edges don't match the cut edges on (4) and (5) — any set built around it in place of (2) leaves a gap. Shape (3) is a quadrilateral similar in outline to (5), but it is shaped for a different corner than the one left once (2) and the other correct quadrilateral are placed, so its cut edges don't match theirs in length. Only (2), (4), and (5) together supply the required triangle-plus-two-quadrilaterals set with every cut edge matched, so they alone reconstruct the equilateral triangle without gap or overlap.