Theory question 112
Progressions(a) The 3rd and 8th terms of an arithmetic progression (A.P.) are −9 and 26 respectively. Find the: (i) common difference; (ii) first term.
(b) In the diagram, PQ||YZ, |XP| = 2 cm, |PY| = 3 cm, |PQ| = 6 cm and the area of ΔXPQ = 24 cm2. Calculate the area of the trapezium PQZY.
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Model answer
(a) a + 2d = −9 and a + 7d = 26. Subtracting: 5d = 35.
(i) d = 7. (ii) a = −9 − 14 = −23.
(b) ΔXPQ ~ ΔXYZ with ratio XP : XY = 2 : 5, so the areas are in the ratio 4 : 25.
Area of ΔXYZ = 24 × 25/4 = 150 cm2.
Area of PQZY = 150 − 24 = 126 cm2.
Explanation
Areas of similar triangles are in the ratio of the squares of their sides.