Find the coordinates of the point which divides the line joining (4, -3) and (-5, 2) in the ratio 3:1.

Understand the Problem

The question is asking to find the coordinates of a point that divides a line segment in a specific ratio, as well as the coordinates of the vertices of a triangle. The problem involves concepts related to geometry and coordinates.

Answer

The coordinates of the point are $ \left( -\frac{11}{4}, \frac{3}{4} \right) $.

Steps to Solve

Identify the coordinates and the ratio

The coordinates of the two points are ( A(4, -3) ) and ( B(-5, 2) ). The ratio in which the line segment is divided is ( 3:1 ).

Use the section formula

The coordinates ((x, y)) of the point dividing the line segment joining two points ( (x_1, y_1) ) and ( (x_2, y_2) ) in the ratio ( m:n ) can be found using the formula:

$$ x = \frac{mx_2 + nx_1}{m+n} $$

$$ y = \frac{my_2 + ny_1}{m+n} $$

Substitute the values into the formula

Using the coordinates ( A(4, -3) ) and ( B(-5, 2) ), the values are ( m = 3 ), ( n = 1 ):

For ( x ): $$ x = \frac{3(-5) + 1(4)}{3 + 1} = \frac{-15 + 4}{4} = \frac{-11}{4} $$
For ( y ): $$ y = \frac{3(2) + 1(-3)}{3 + 1} = \frac{6 - 3}{4} = \frac{3}{4} $$

Final coordinates

The coordinates of the point that divides the line segment are:

$$ \left( -\frac{11}{4}, \frac{3}{4} \right) $$

The coordinates of the point are ( \left( -\frac{11}{4}, \frac{3}{4} \right) ).

More Information

This problem utilizes the section formula from coordinate geometry, which helps determine the coordinates of a point dividing a line segment in a specified ratio. Understanding the section formula is useful in many areas of math involving geometry and coordinate systems.

Tips

Confusing the ratio direction: Ensure the values of ( m ) and ( n ) correctly correspond to the points ( A ) and ( B ).
Incorrectly applying the formula: Double-check the computations for both ( x ) and ( y ) to avoid arithmetic errors.

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