Section Formula — Assertion-Reason Type Questions

Assertion (A): The point P(3, -1) divides the line segment joining the points A(1, -3) and B(6, 2) internally in the ratio 2 : 3.

Reason (R): The coordinates of the point which divides the line segment joining the points A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂ are $\Big(\dfrac{m_1x_2 + m_2x_1}{m_1 + m_2}, \dfrac{m_1y_2 + m_2y_1}{m_1 + m_2}\Big)$.

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).

Answer

By section-formula,

The coordinates of the point which divides the line segment joining the points A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂ are

$\Big(\dfrac{m_1x_2 + m_2x_1}{m_1 + m_2}, \dfrac{m_1y_2 + m_2y_1}{m_1 + m_2}\Big)$.

So, reason (R) is true.

Let us find the point P which divides the line segment joining A(1, -3) and B(6, 2) in the ratio 2 : 3, by section-formula.

$\Rightarrow \text{P} = \Big(\dfrac{2 \times 6 + 3 \times 1}{2 + 3}, \dfrac{2 \times 2 + 3 \times (-3)}{2 + 3}\Big)\\[1em] = \Big(\dfrac{12 + 3}{5}, \dfrac{4 - 9}{5}\Big)\\[1em] = \Big(\dfrac{15}{5}, \dfrac{-5}{5}\Big)\\[1em] = (3, -1)$

So, assertion (A) is true.

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

Hence, option 3 is the correct option.

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).

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 = $\dfrac{1}{2}$ BC.

By distance formula,

Distance between two points = $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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

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.

Assertion (A): The point (-5, 0) lies on the x-axis.

Reason (R): The y-coordinates of a point on x-axis is zero.

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).

Answer

We know that,

A point lies on the x-axis if its y-coordinate is 0.

So, reason (R) is true.

Here, the point is (−5, 0), so the y-coordinate is 0.

So, assertion (A) is true.

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

Hence, option 3 is the correct option.