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If the point P (2, 1) lies on the line segment joining points A (4, 2) and B (8, 4), then
(A) AP = 1/3 AB
(B) AP = PB
(C) PB = 1/3 AB
(D) AP = 1/2 AB

Solution:

We know that

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

(x, y) = \([\frac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}},\frac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}]\)

Consider P to divide AB in the ratio k:1

Let us substitute the values (x₁, y₁) = (4, 2) and (x₂, y₂) = (8, 4)

\(P=[\frac{k(8)+1(4)}{k+1},\frac{k(4)+1(2)}{k+1}]\)

It is given that P = (2, 1)

\(P=[\frac{8k+4}{k+1},\frac{4k+2}{k+1}]\) = (2, 1)

Let us compare the x coordinates

(8k + 4)/ (k + 1) = 2

8k + 4 = 2k + 2

6k = -2

Divide both sides by -2

k = -1/3

As k value is negative, P divides AB in the ratio 1:3 externally

AP/PB = 1/3

AP = 1/2 AB

Therefore, the line segment joining points A (4, 2) and B (8, 4), then AP = 1/2 AB.

✦ Try This: If the point P (3, 1) lies on the line segment joining points A (2, 4) and B (6, 3), then

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 7

NCERT Exemplar Class 10 Maths Exercise 7.1 Problem 12

If the point P (2, 1) lies on the line segment joining points A (4, 2) and B (8, 4), then (A) AP = 1/3 AB, (B) AP = PB, (C) PB = 1/3 AB, (D) AP = 1/2 AB

Summary:

If the point P (2, 1) lies on the line segment joining points A (4, 2) and B (8, 4), then AP = 1/2 AB

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