point c (2,1) is the midpoint of line segment ab. Find the coordinates of points a and b if they are both on axes. explain and justify your reasoning

By similar triangles, if (2,1) is the mid-point of line ab,

then the y-intercept is 2*1=2 at (0,2), and the x-intercept is 2*2=4 at (4,0)

To find the coordinates of points A and B if point C is the midpoint of line segment AB, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint between two points (x₁, y₁) and (x₂, y₂) are given by:

Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

In this case, we know point C is the midpoint, which is (2, 1). Since points A and B are on the axes, either their x-coordinate or y-coordinate will be zero.

Let's consider the case where point A lies on the x-axis (A(x, 0)). Since C is the midpoint, the x-coordinate of A will be twice the x-coordinate of C. Therefore, x = 2 * 2 = 4. So the coordinates of point A are (4, 0).

Now, to find the coordinates of point B, we subtract the x-coordinate of C from twice the x-coordinate of A to get the x-coordinate of B. B(x, 0) → x = 2 * 4 - 2 = 6. So the coordinates of point B are (6, 0).

Therefore, the coordinates of points A and B when C(2, 1) is the midpoint of line segment AB, and both points A and B are on the axes, are A(4, 0) and B(6, 0).

We can also justify this reasoning by considering that if C is the midpoint of line segment AB, the distance between A and C should be equal to the distance between C and B. By calculating these distances, we can confirm that point A is (4, 0) and point B is (6, 0).