Theory question 111
Triangles and Quadrilaterals(a) In the diagram, ∠PTQ = ∠PSR = 90°, |PQ| = 10cm, |PS| = 14.4cm and |TQ| = 6cm. Calculate the area of quadrilateral QRST.
(b) Two opposite sides of a square are each decreased by 10% while the other two are each increased by 15% to form a rectangle. Find the ratio of the area of the rectangle to that of the square.
Show the model answer
Model answer
(a) |PT| = √(102 − 62) = 8 cm. ΔPTQ ~ ΔPSR with scale factor 14.4/8 = 1.8, so |SR| = 6 × 1.8 = 10.8 cm.
Area QRST = ½ × 14.4 × 10.8 − ½ × 8 × 6 = 77.76 − 24 = 53.76 cm2.
(b) Side s: rectangle = 0.9s × 1.15s = 1.035s2. Ratio = 1.035 : 1 = 207 : 200.
Explanation
TQ and SR are both perpendicular to PS, so the triangles are similar.