The coordinates of point T are given. The midpoint of segment ST is (5, -8). Find the coordinates of
point S.
1. T(0,4)
2.T(-2,8)
Let's solve the problem step by step.
For 1. T(0,4):
To find the coordinates of point S, we need to use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment are the average of the coordinates of the two endpoints.
Given that the midpoint of segment ST is (5, -8) and the coordinates of point T are (0,4), we can use the midpoint formula to find the coordinates of point S.
Midpoint formula:
Midpoint = ( (x1 + x2)/2, (y1 + y2)/2 )
Let's substitute the known values into the formula:
(5, -8) = ( (0 + x2)/2, (4 + y2)/2 )
Simplifying the equation, we have:
(5, -8) = ( x2/2, (4 + y2)/2 )
To solve for x2, multiply both sides of the equation by 2:
(10, -16) = ( x2, 4 + y2 )
Therefore, the coordinates of point S are (10, -16).
For 2. T(-2,8):
Using the same process as above, we substitute the known values into the midpoint formula:
(5, -8) = ( (-2 + x2)/2, (8 + y2)/2 )
Simplifying the equation, we have:
(5, -8) = ( (x2 - 2)/2, (8 + y2)/2 )
To solve for x2, multiply both sides of the equation by 2:
(10, -16) = ( x2 - 2, 8 + y2 )
Therefore, the coordinates of point S are (10, -16).
In both cases, the coordinates of point S are (10, -16).
To find the coordinates of point S, we will use the midpoint formula.
The midpoint formula states that the coordinates of the midpoint (M) of a line segment with endpoints (x1, y1) and (x2, y2) can be found by taking the average of the x-coordinates and the average of the y-coordinates:
Midpoint (M): ((x1 + x2)/2, (y1 + y2)/2)
Let's use this formula to find the coordinates of point S for each given scenario:
1. T(0, 4):
Given: Midpoint = (5, -8)
Let's plug in the values into the formula:
(5, -8) = ((0 + x2)/2, (4 + y2)/2)
To find the x-coordinate of point S, we can solve for x2:
2 * 5 = 0 + x2
10 = x2
To find the y-coordinate of point S, we can solve for y2:
2 * (-8) = 4 + y2
-16 = 4 + y2
-20 = y2
Therefore, the coordinates of point S are S(10, -20).
2. T(-2, 8):
Given: Midpoint = (5, -8)
Let's plug in the values into the formula:
(5, -8) = ((-2 + x2)/2, (8 + y2)/2)
To find the x-coordinate of point S, we can solve for x2:
2 * 5 = -2 + x2
10 = -2 + x2
12 = x2
To find the y-coordinate of point S, we can solve for y2:
2 * (-8) = 8 + y2
-16 = 8 + y2
-24 = y2
Therefore, the coordinates of point S are S(12, -24).
The midpoint M = (S+T)/2 in both x and y coordinates. So
#1. (S+(0,4))/2 = (5,-8)
S+(0,4) = (10,-16)
S = (10,-16)-(0,4) = (10,-20)
or,
S is as far from M as T is, but in the opposite direction
0 = 5-5, so you want 5+5=10
4 = -8+12, so you want -8-12 = -20