If point P divides the segment joining A(1, 2) and B(5, 6) in the ratio 1:3, wha

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Question: If point P divides the segment joining A(1, 2) and B(5, 6) in the ratio 1:3, what are the coordinates of P?

Options:

Correct Answer: (3, 4)

Solution:

Using the section formula: P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) = ((1*5 + 3*1)/(1+3), (1*6 + 3*2)/(1+3)) = (3, 4).

If point P divides the segment joining A(1, 2) and B(5, 6) in the ratio 1:3, what are the coordinates of P?

If point P divides the segment joining A(1, 2) and B(5, 6) in the ratio 1:3, what are the coordinates of P?

Step 1: Identify the coordinates of points A and B. A is at (1, 2) and B is at (5, 6).
Step 2: Understand that point P divides the line segment AB in the ratio 1:3. This means for every 1 part from A, there are 3 parts towards B.
Step 3: Use the section formula to find the coordinates of point P. The formula is P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)), where m and n are the parts of the ratio.
Step 4: Assign m = 1 and n = 3. The coordinates of A are (x1, y1) = (1, 2) and the coordinates of B are (x2, y2) = (5, 6).
Step 5: Substitute the values into the formula for the x-coordinate: P_x = ((1*5 + 3*1)/(1+3)). Calculate this: P_x = (5 + 3)/(4) = 8/4 = 2.
Step 6: Substitute the values into the formula for the y-coordinate: P_y = ((1*6 + 3*2)/(1+3)). Calculate this: P_y = (6 + 6)/(4) = 12/4 = 3.
Step 7: Combine the x and y coordinates to find point P. So, P = (2, 3).

Section Formula – The section formula is used to find the coordinates of a point that divides a line segment in a given ratio.
Ratio Division – Understanding how to apply the ratio to the coordinates of the endpoints to find the coordinates of the dividing point.