The mid-point of the straight line joining A(4,-3), B(6,-2) is located at C.
(a) Calculate the coordinates of C.
(b) Verify your result by calculating the distances AC and BC and showing they are equal.
midpoint between (x1, y1) and (x2, y2)
= ( (x1+x2)/2 , (y1 + y2)/2 )
a)
so C = ( (4+6)/2 , (-3-2)/2 )
= (5 , -5/2)
b) use the method I just showed you in the previous post to find the two length,
show them equal.
To calculate the coordinates of the midpoint C, we can use the midpoint formula, which states that the mid-point of a line segment joining two points A(x1, y1) and B(x2, y2) is given by the coordinates:
\[C(\frac{{x1+x2}}{2}, \frac{{y1+y2}}{2})\]
(a) Applying the formula, we have:
The x-coordinate of C = \(\frac{{4+6}}{2} = 5\)
The y-coordinate of C = \(\frac{{-3+(-2)}}{2} = \frac{{-5}}{2} = -2.5\)
Therefore, the coordinates of C are (5, -2.5).
(b) Now, let's verify the result by calculating the distances AC and BC to demonstrate that they are equal.
To calculate the distance between two points in a coordinate plane, we can use the distance formula, which states that the distance between two points A(x1, y1) and B(x2, y2) is given by the formula:
\[d = \sqrt{{(x2-x1)^2 + (y2-y1)^2}}\]
Calculating the distance AC:
AC = \(\sqrt{{(5-4)^2 + (-2.5-(-3))^2}}\)
= \(\sqrt{{1^2 + 0.5^2}}\)
= \(\sqrt{{1 + 0.25}}\)
= \(\sqrt{{1.25}}\)
Calculating the distance BC:
BC = \(\sqrt{{(6-5)^2 + (-2-(-2.5))^2}}\)
= \(\sqrt{{1^2 + 0.5^2}}\)
= \(\sqrt{{1 + 0.25}}\)
= \(\sqrt{{1.25}}\)
As we can see, the distances AC and BC are both equal to \(\sqrt{{1.25}}\), which confirms that the point C is indeed the midpoint of the line segment AB.