An expert question on perimeter and area for P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P4 to P6 papers; harder ones work out areas from the ratios between triangles' areas.
The question
In the figure, K is a point on the diagonal PR of a rectangular flower bed PQRS.
A gardener lays two paths through K: one parallel to PS, from M on PQ to N on SR; the other parallel to PQ, from U on PS to V on QR.
Triangle PMK has an area of 12 cm² and triangle KRN 27 cm².
What is the area of rectangle PQRS, in cm²?
Show the hints (3)
💡Hint 1: Which two small rectangles does the diagonal PR cross? How does it cut each of them?
💡Hint 2: Compare triangles PQR and PRS: how do the two small rectangles off the diagonal compare?
💡Hint 3: Of the four small rectangles, top-left times bottom-right equals top-right times bottom-left.
Worked solution
Have a go first, then read the solution step by step.
1.The diagonal PR cuts rectangle PMKU into two equal triangles, so PMKU = 12 × 2 = 24 cm²; likewise KVRN = 27 × 2 = 54 cm².
2.Triangles PQR and PRS are equal. PQR = PMK + MQVK + KVR and PRS = PUK + UKNS + KRN, with PMK = PUK and KVR = KRN, so UKNS = MQVK.
3.Now the sides of the four small rectangles: PMKU = PM × MK, KVRN = KV × VR, MQVK = MQ × MK, UKNS = UK × KN. Since UK = PM, MQ = KV and KN = VR, PMKU × KVRN = MQVK × UKNS (opposite corners multiply to the same).
4.MQVK and UKNS are equal, so MQVK × MQVK = 24 × 54 = 1296; 36 × 36 = 1296, so MQVK = 36 cm².
5.Rectangle PQRS = 24 + 54 + 36 × 2 = 150 cm².
⚠️Common mistake: taking MQVK as the average of PMKU and KVRN. The rectangles off the diagonal come from 'opposite rectangles multiply to the same'.
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