Question P5–P6 Overlap and cancel
Two identical right-angled triangles overlap. Their perpendicular sides are 12 cm and 18 cm; the shaded parts total 132 cm². Find the common unshaded area.
Answer
42 cm². Each triangle has area 1/2 × 12 × 18 = 108 cm². Since the identical triangles contain the same unshaded overlap, their shaded remainders are equal: 132 ÷ 2 = 66 cm² each. Hence the unshaded area is 108 − 66 = 42 cm². Check: each triangle is 66 + 42 = 108 cm², and the two shaded remainders total 66 + 66 = 132 cm², exactly as given.
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Two shapes share a region, and the shared part cancels out.
In this one. Equal whole triangles with the same overlap must leave equal shaded remainders, so split the total shaded area equally.
Shapes are joined, stacked, or have a piece cut off, and the question compares a perimeter or length before and after, rather than asking for an area from scratch.
Children assume perimeter changes the same way area does. Joining shapes or cutting off a corner can leave the perimeter unchanged, or change it by a different amount from the area.
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