Theory question 108
Progressions(a) If (3 − x), 6, (7 − 5x) are consecutive terms of a geometric progression (G.P.) with constant ratio r > 0; find the: (i) values of x; (ii) constant ratio.
(b) In the diagram, |AB| = 3 cm, |BC| = 4 cm, |CD| = 6 cm and |DA| = 7cm. Calculate ∠ADC, correct to the nearest degree.
Show the model answer
Model answer
(a)(i) 62 = (3 − x)(7 − 5x) → 36 = 21 − 22x + 5x2 → 5x2 − 22x − 15 = 0 → (5x + 3)(x − 5) = 0 → x = −3/5 or 5.
x = 5 gives −2, 6, −18 (ratio −3, not allowed). So x = −3/5.
(ii) Terms 18/5, 6, 10: r = 6 ÷ 18/5 = 5/3.
(b) ∠ABC = 90°, so |AC| = √(32 + 42) = 5 cm.
In ΔACD: cos ∠ADC = (72 + 62 − 52)/(2 × 7 × 6) = 60/84 ≈ 0.714, so ∠ADC ≈ 44°.
Explanation
In a GP the middle term squared equals the product of its neighbours.