(a) If (3 - x), 6, (7 - 5x) are consecutive terms of a geometric progression (GP) 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| = 7 cm. Calculate <ADC, correct to the nearest degree.

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(a) If (3 - x), 6, (7 - 5x) are consecutive terms of a geometric progression (GP) with ...

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(a) If (3 - x), 6, (7 - 5x) are consecutive terms of a geometric progression (GP) 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| = 7 cm. Calculate <ADC, correct to the nearest degree.

Explanation

(a) (i) (3 - x), 6, (7 - 5x).
First term = a = (3 - x)
common ratio =

\frac{6}{3 - x} = \frac{7 - 5 x}{6}

36 = (3 - x) (7 - 5 x)

36 = 21 - 15 x - 7 x + 5 x^{2}

36 = 21 - 22 x + 5 x^{2}

5 x^{2} - 22 x + 21 - 36 = 0

5 x^{2} - 22 x - 15 = 0

5 x^{2} - 25 x + 3 x - 15 = 0 ⟹ 5 x (x - 5) + 3 (x - 5) = 0

(x - 5) (5 x + 3) = 0

x = - \frac{3}{5} or = 5

(ii) Constant ratio =

\frac{6}{3 - x} = \frac{7 - 5 x}{6}

Using x = 5,

r = \frac{7 - 5 (5)}{6} = \frac{- 18}{6}

- 3

(b) Considering

Δ

AOC

b^{2} = 4^{2} + 3^{2}

b^{2} = 16 + 9 = 25

b = \sqrt{25} = 5 c m

Considering

Δ

ACD,
Using cosine rule,

\cos D = \frac{a^{2} + b^{2} - c^{2}}{2 a b} = \frac{a^{2} + c^{2} - b^{2}}{2 a c}

\frac{6^{2} + 7^{2} - 5^{2}}{2 (6) (7)}

\frac{36 + 49 - 25}{84}

\cos D = \frac{60}{84}

\cos D = 0.714

D = \cos^{- 1} (0.714)

D = 44.4 ° ≊ 44 °

(to the nearest degree)

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