In figure if $A D = 6 c m, D B = 9 c m, A E = 8 c m$ and $E C = 12 c m$ and $∠ A D E = 48^{o}$ . Find $∠ A B C$ .

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Solution

In $△ A D E, a n d △ A B C$

$A D = 6, A B = A D + D B = 6 + 9 = 15 ⟹ \frac{A D}{A B} = \frac{6}{15} = \frac{2}{5}$

$A E = 8, A C = A E + E C = 8 + 12 = 20 ⟹ \frac{A E}{A C} = \frac{8}{20} = \frac{2}{5}$

$\Rightarrow \frac{A D}{A B} = \frac{A E}{A C}, ∠ D A E = ∠ B A C$ (common angle)

As sides are proportional and an angle is equal

$\Rightarrow △ A B C \sim △ A D E$

$∴ ∠ A B C = ∠ A D E = 48^{\circ}$

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Q. In a ∆ABC, D and E are points on the sides AB and AC respectively such that DE || BC.

(i) If AD = 6 cm, DB = 9 cm and AE = 8 cm, find AC.
(ii) If

\frac{AD}{DB} = \frac{3}{4}

and AC = 15 cm, find AE.
(iii) If

\frac{AD}{DB} = \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, find 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 = 9 cm, find AE.
(viii) If

\frac{AD}{BD} = \frac{4}{5}

and EC = 2.5 cm, find AE.
(ix) If AD = 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, BD = 3x − 1 and CE = 5x − 3. find the volume x.
(xii) If AD = 2.5 cm, BD = 3.0 cm and AE = 3.75 cm find the length of AC.

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.