Today’s problem — free all day, every step. Every day has one.
Garden PQRS — the path and the shaded triangles
A rectangular garden PQRS is divided into two rectangular beds by a straight path UV, where U is on PQ and V is on SR (so PUVS and UQRV are the two beds). H is a sprinkler point on PS, D is a sprinkler point on QR, and M is the midpoint of UV. The perimeter of PQRS is 84 m and PS : PQ = 3 : 4. Find (a) PS, and (b) the total area of the shaded triangles HMV and UMD.
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(a) PS = 18 m. (b) Total shaded area = 108 m².
The pattern behind it: Overlap and cancel
Two shapes share a region, and the shared part cancels out.
In this one. The two triangles have equal 9 m bases on UV, while their perpendicular heights are PU and UQ, which join to make the full width PQ; therefore the split point U does not affect their total area.
How to spot it
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.
Common mistake
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.
Try one like it
Standard Area of triangle Two identical right-angled triangles overlap. Their perpendicular sides are 12 cm and 18 cm; the shaded parts total 132 cm². Find …The 33 problem patterns
All 33 patterns, one to a page — the 19 behind Primary 5 and 6 word problems, and the skills the younger years are built on. The bar model, how to spot it in the question, and the mistake children make. A booklet you can print. Free.
A PDF, and a note when new solutions go up.