D and E are points on the sides AB and AC respectively of a $Δ A B C$ such that DE || BC
Find the value of x, when

(i) AD = x cm, DB = (x - 2) cm,

AE = (x + 2) cm and EC = (x - 1) cm.

(ii) AD = 4 cm, DB = (x - 4) cm, AE = 8 cm and EC = (3x - 19) cm.

(iii) AD = (7x - 4) cm, AE = (5x - 2) cm, DB = (3x + 4 ) cm and EC = 3x cm.

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Solution

(i)It is given that AD = x, DB = x – 2, AE = x + 2 and EC = x – 1

We have to find the value of x

So, $\frac{A D}{B D} = \frac{A E}{C E}$ (using Thales Theorem)

Then, $\frac{x}{x - 2} = \frac{x + 2}{x - 1}$

$x (x - 1) = (x - 2) (x + 2)$

$x^{2} - x - x^{2} + 4 = 0$

$x = 4$

(ii)It is given that AD = 4 cm, AE = 8 cm, DB = x – 4 and EC = 3x – 19

We have to find x,

So, $\frac{A D}{B D} = \frac{A E}{C E}$ (using Thales Theorem)

Then, $\frac{4}{x - 4} = \frac{8}{3 x - 19}$

4(3x – 19) = 8(x – 4)

12x – 76 = 8(x – 4)

12x – 8x = – 32 + 76

4x = 44 cm

x = 11 cm

(iii) It is given that AD = (7x - 4) cm, AE = (5x - 2) cm, DB = (3x + 4 ) cm and EC = 3x cm

We have to find x,

So, $\frac{A D}{B D} = \frac{A E}{C E}$ (using Thales Theorem)

Then, $\frac{7 x - 4}{3 x + 4} = \frac{5 x - 2}{3 x}$

$3 x (7 x - 4) = (5 x - 2) (3 x + 4)$

$21 x^{2} - 12 x = 15 x^{2} + 20 x - 6 x - 8$

$21 x^{2} - 15 x^{2} - 12 x - 14 x + 8 = 0$

$6 x^{2} - 26 x + 8 = 0$

$3 x^{2} - 13 x + 4 = 0$

a = 3, b = -13 and c = 4

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} = \frac{- (- 13) \pm \sqrt{(- 13)^{2} - 4 \times 3 \times 4}}{2 \times 3} = \frac{13 \pm \sqrt{169 - 48}}{6} = \frac{13 \pm \sqrt{121}}{6} = \frac{13 \pm 11}{6} = \frac{13 + 11}{6} o r \frac{13 - 11}{6} = \frac{24}{6} o r \frac{2}{6} = 4 o r \frac{1}{3}$

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Similar questions

Q. In ∆ABC, DE ∥ BC so that AD = (7x − 4) cm, AE = (5x − 2) cm, DB = (3x + 4) cm and EC = 3x cm. Then, we have:

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Q. In the given figure, DE ∥ BC such that AD = x cm, DB = (3x + 4) cm, AE = (x + 3) cm and EC = (3x + 19) cm. Find the value of x.

In a \Delta ABC, D and E are points on the sides AB and AC respectively such that DE||BC.
(i) If AD = 6cm, DB = 9 cm and AE = 8 cm, find AC.
(ii) If $\frac{A D}{D B} = \frac{3}{4}$ and AC = 15 cm, find AE.
(iii) If $\frac{A D}{D B} = \frac{2}{3}$ and AC = 18 cm, find AE.
(iv) If AD = 4, AE=8, DB =x-4, and EC = 3x - 19 find x
(v) If AD = 8 cm, AB=12 cm and AE = 12 cm, finde CE.
(vi) If ad = 4 CM, db = 4.5 cm and AE = 8 cm, find AC
(vii) If AD = 2 cm, AB = 6 cm and AC = 9cm, find AE.
(viii) If $\frac{A D}{B D} = \frac{4}{5}$ and 3C = 2.5 cm, find AE.
(ix) If AC=x, DB=x-2, AE=x+2 and EC=x-1, find the value of x.
(x) If AD= 8x -7. DB= 5x-3, AE=4x-3 and EC= (3x-1), find the value of x.
(xi) If AD= 4x-3, AE=8x -7, BC=3x-1 and CE=5x-3, find the value x.
(xii) If AD=2.5 cm, BD= =3.0 cm and AE=3.75 cm, find the length of AC.

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