Find the coordinates of the point which divides the line segment … - Vidyadip

Question

Find the coordinates of the point which divides the line segment joining the points (4,-3) and (8, 5) in the ratio 3:1 internally.

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Answer

Let the two given points be $A(4,-3)$ and $B(8,5)$.
Let the ratio of division be $m_1: m_2=3: 1$.
The coordinates of the point $P(x, y)$ that divides the line segment AB internally in the ratio $m_1: m_2$ are given by the formula:
$P(x, y)=\left(\frac{m_1 x_2+m_2 x_1}{m_1+m_2}, \frac{m_1 y_2+m_2 y_1}{m_1+m_2}\right)$
Substitute the given values into the formula:
$\begin{array}{l}x_1=4, y_1=-3 \\ x_2=8, y_2=5 \\ m_1=3, m_2=1\end{array}$
For the x-coordinate:
$x=\frac{(3)(8)+(1)(4)}{3+1}=\frac{24+4}{4}=\frac{28}{4}=7$
For the y-coordinate:
$y=\frac{(3)(5)+(1)(-3)}{3+1}=\frac{15-3}{4}=\frac{12}{4}=3$
Therefore, the coordinates of the point that divides the line segment are (7, 3).

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