If point P is at (3, 5) and point Q is at (9, 1), what is the coordinates of poi

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Question: If point P is at (3, 5) and point Q is at (9, 1), what is the coordinates of point R that divides PQ in the ratio 2:3?

Options:

Correct Answer: (5.4, 3.6)

Solution:

Using the section formula: R = ((2*9 + 3*3)/(2+3), (2*1 + 3*5)/(2+3)) = (5.4, 3.6).

If point P is at (3, 5) and point Q is at (9, 1), what is the coordinates of point R that divides PQ in the ratio 2:3?

If point P is at (3, 5) and point Q is at (9, 1), what is the coordinates of point R that divides PQ in the ratio 2:3?

Step 1: Identify the coordinates of points P and Q. Point P is at (3, 5) and point Q is at (9, 1).
Step 2: Understand that we need to find point R that divides the line segment PQ in the ratio 2:3.
Step 3: Use the section formula, which is R = ((m*x2 + n*x1)/(m+n), (m*y2 + n*y1)/(m+n)), where m and n are the parts of the ratio.
Step 4: In our case, m = 2 and n = 3. The coordinates of point P (x1, y1) are (3, 5) and the coordinates of point Q (x2, y2) are (9, 1).
Step 5: Substitute the values into the section formula for the x-coordinate: R_x = (2*9 + 3*3)/(2+3).
Step 6: Calculate R_x: R_x = (18 + 9)/(5) = 27/5 = 5.4.
Step 7: Now substitute the values into the section formula for the y-coordinate: R_y = (2*1 + 3*5)/(2+3).
Step 8: Calculate R_y: R_y = (2 + 15)/(5) = 17/5 = 3.4.
Step 9: Combine the results to find the coordinates of point R: R = (5.4, 3.4).

Section Formula – The section formula is used to find the coordinates of a point that divides a line segment into a given ratio.
Coordinate Geometry – Understanding the Cartesian coordinate system and how to manipulate coordinates is essential for solving problems involving points in a plane.
Ratios – The concept of ratios is crucial for determining how to weight the coordinates of the points being divided.