If point C(1, 1) divides the line segment joining points A(0, 0) and B(4, 4) in

If point C(1, 1) divides the line segment joining points A(0, 0) and B(4, 4) in the ratio k:1, what is the value of k?

If point C(1, 1) divides the line segment joining points A(0, 0) and B(4, 4) in the ratio k:1, what is the value of k?

Step 1: Identify the coordinates of points A and B. A is (0, 0) and B is (4, 4).
Step 2: Identify the coordinates of point C, which is (1, 1).
Step 3: Understand that point C divides the line segment AB in the ratio k:1.
Step 4: Use the section formula for the x-coordinate: Cx = (kx2 + x1) / (k + 1). Here, Cx = 1, x1 = 0, and x2 = 4.
Step 5: Substitute the values into the formula: 1 = (k * 4 + 0) / (k + 1).
Step 6: Simplify the equation: 1 = (4k) / (k + 1).
Step 7: Cross-multiply to eliminate the fraction: 1 * (k + 1) = 4k.
Step 8: Expand the left side: k + 1 = 4k.
Step 9: Rearrange the equation to isolate k: 1 = 4k - k, which simplifies to 1 = 3k.
Step 10: Solve for k by dividing both sides by 3: k = 1/3.
Step 11: Verify the solution by checking if C(1, 1) divides AB in the ratio 1/3:1.

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 express the division of a line segment in terms of a ratio and how it relates to the coordinates of the points.