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Equal Intercept Theorem

In a Δ ABC,...

Question

In a $Δ A B C$ , $D$ and $E$ are points on the sides $A B$ and $A C$ respectively such that $D E | | B C$ .
If $A D = 4 x - 3, A E = 8 x - 7, B D = 3 x - 1$ and $C E = 5 x - 3$ , find the value of $x$ .

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Solution

$\begin{matrix} I n Δ A D E & Δ A B C ∠ A = ∠ A & ∠ A D E = ∠ A B C ∴ Δ A D E \sim Δ A B C [A A - s i m i l a r i t y] \frac{A D}{A B} = \frac{A E}{A C} \frac{8 x - 7}{8 x - 7 + 5 x - 3} = \frac{4 x - 3}{4 x - 3 + 3 x - 1} \Rightarrow (8 x - 7) (7 x - 4) = (4 x - 3) (13 x - 10) \Rightarrow 56 x^{2} - 32 x - 49 x + 28 = 52 x^{2} - 40 x - 39 x + 30 \Rightarrow 56 x^{2} - 81 x + 28 - 52 x^{2} + 79 x - 30 = 0 \Rightarrow 4 x^{2} - 2 x - 2 = 0 \Rightarrow 2 x^{2} - x - 1 = 0 \Rightarrow 2 x^{2} - 2 x + x - 1 = 0 \Rightarrow 2 x^{2} - 2 x + x - 1 = 0 \Rightarrow 2 x (x - 1) + 1 (x - 1) = 0 \Rightarrow (2 x + 1) (x - 1) = 0 ∴ 2 x + 1 = 0 a n d x - 1 = 0 ∴ x = \frac{- 1}{2} a n d x = 1 s i n c e l e n g t h c a n n o t b e - v e ∴ x = 1. \end{matrix}$

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ABC, D and E are points on the sides AB and AC respectively such that DE || BC. If AD =

4 x - 3

3 x - 1

8 x - 7

5 x - 3

x

Δ A B C

D

E

are points on the sides

A B

A C

respectively such that

D E | | B C

A D = 8 x - 7, D B = 5 x - 3, A E = 4 x - 3

E C = (3 x - 1)

, find the value of

x

Δ A B C, D a n d E

are points on the sides

A B

A C

respectively such that

D E ∥ B C

A D = 4 x - 3, A E = 8 x - 7, B D = 3 x - 1 a n d C E = 5 x - 3

, find the value of

x

.

Δ A B C

D

E

are points on the sides

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A C

respectively such that

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A D = 8 c m, A B = 12

A E = 12 c m

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Q. In a ∆ABC, D and E are points on AB and AC respectively such that DE || BC. If AD = 2.4 cm, AE = 3.2 cm, DE = 2 cm and BC = 5 cm, find BD and CE.

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