Assertion (A): In the given figure, DE ? BC, so that AD = (4x – 3) cm, AE = (8x – 7) cm,

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Solution

Q.
Assertion (A): In the given figure, DE || BC, so that AD = (4x – 3) cm, AE = (8x – 7) cm, BD = (3x – 1) cm and CE = (5x – 3) cm, then the value of x is 1.

Reason (R): In triangle ABC, if DE || BC, then $\frac{A D}{B D} = \frac{A E}{C E}$

1 Both A and R are true, and R is the correct explanation of A.

2 Both A and R are true, but R is not the correct explanation of A

3 A is true, but R is false.

4 A is false, but R is true.

Ans.
(1)

$S i n c e, i n ∆ A B C, D E | | B C \frac{A D}{D B} = \frac{A E}{E C} [By BPT] \frac{4 x - 3}{3 x - 1} = \frac{8 x - 7}{5 x - 3} \Rightarrow (4 x - 3) (5 x - 3) = (8 x - 7) (3 x - 1) \Rightarrow 20 x^{2} - 12 x - 15 x + 9 = 24 x^{2} - 8 x - 21 x + 7 \Rightarrow 20 x^{2} - 27 x + 9 = 24 x^{2} - 29 x + 7 \Rightarrow 4 x^{2} - 2 x - 2 = 0 \Rightarrow 2 x^{2} - x - 1 = 0 \Rightarrow 2 x^{2} - 2 x + x - 1 = 0 \Rightarrow 2 x (x - 1) + 1 (x - 1) = 0 \Rightarrow (x - 1) (2 x + 1) = 0 \Rightarrow x = 1 [∵ x \neq - \frac{1}{2}]$

Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

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