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Mathematics

Assertion (A): If the coordinates of the mid-points of the sides AB and AC of Δ ABC are D(3, 5) and E(-3, 5) respectively, then BC = 12 units.

Reason (R): The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it.

  1. Assertion (A) is true, Reason (R) is false.

  2. Assertion (A) is false, Reason (R) is true.

  3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

  4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Section Formula

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Answer

By mid-point theorem,

The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it."

So, reason (R) is true.

Given,

D(3, 5) is mid-points of AB and E(-3, 5) is mid-points of AC. By midpoint Theorem:

DE || BC and DE = 12\dfrac{1}{2} BC.

By distance formula,

Distance between two points = (x2−x1)2+(y2−y1)2\sqrt{(x2 - x1)^2 + (y2 - y1)^2}

DE=((−3)−3)2+(5−5)2=(−6)2+02=36=6 unitsDE = \sqrt{((-3) - 3)^2 + (5 - 5)^2}\\[1em] = \sqrt{(-6)^2 + 0^2}\\[1em] = \sqrt{36}\\[1em] = \text{6 units}

BC = 2 x DE = 2 x 6 = 12 units.

So, assertion (A) is true.

Thus, both A and R are correct, and R clearly explains A.

Hence, option 3 is the correct option.

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