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

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Question: 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?

Options:

Correct Answer: 2

Solution:

Using the section formula, C = ((kx2 + x1)/(k+1), (ky2 + y1)/(k+1)). Setting C(1, 1) gives k = 1.

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.