Suppose the cordinate of P is 2, PQ = 8, and PR = 12. What are the possible cordinates of the midpoint of the given segment?
PQ
PR
QR
To find the possible coordinates of the midpoint of segment PQ, PR, and QR, we need to 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) are as follows:
M = ((x1 + x2) / 2, (y1 + y2) / 2)
Let's apply this formula to each line segment one by one:
1. PQ:
Given P(2, _) and Q(_, _), where PQ = 8.
Let's assume the coordinates of Q are (x, y). Using the distance formula, we can find the equation:
√((x - 2)^2 + (y - _)^2) = 8
Simplifying, we get:
(x - 2)^2 + (y - _)^2 = 64
Since we don't know the value of y for point Q, let's rewrite the equation as:
(x - 2)^2 + y^2 = 64
Now, let's find the value of x when y = 0:
(x - 2)^2 + 0^2 = 64
(x - 2)^2 = 64
Taking the square root of both sides, we have:
x - 2 = ±8
x = 2 ± 8
Thus, the two possible coordinates for point Q are (10, 0) and (-6, 0).
Therefore, the possible coordinates for the midpoint of segment PQ are ((2 + 10) / 2, (0 + 0) / 2) = (6, 0) and ((2 + -6) / 2, (0 + 0) / 2) = (-2, 0).
2. PR:
Given P(2, _) and R(_, _), where PR = 12.
Let's assume the coordinates of R are (x, y). Using the distance formula, we can find the equation:
√((x - 2)^2 + (y - _)^2) = 12
Simplifying, we get:
(x - 2)^2 + (y - _)^2 = 144
Since we don't know the value of y for point R, let's rewrite the equation as:
(x - 2)^2 + y^2 = 144
Now, let's find the value of x when y = 0:
(x - 2)^2 + 0^2 = 144
(x - 2)^2 = 144
Taking the square root of both sides, we have:
x - 2 = ±12
x = 2 ± 12
Thus, the two possible coordinates for point R are (14, 0) and (-10, 0).
Therefore, the possible coordinates for the midpoint of segment PR are ((2 + 14) / 2, (0 + 0) / 2) = (8, 0) and ((2 + -10) / 2, (0 + 0) / 2) = (-4, 0).
3. QR:
Given Q(_, _) and R(_, _), where QR = 8.
Let's assume the coordinates of Q are (x1, y1) and the coordinates of R are (x2, y2). The midpoint coordinates (M) can be found using the midpoint formula:
M = ((x1 + x2) / 2, (y1 + y2) / 2)
Since we don't have any specific information about the coordinates of Q and R, we cannot determine the exact values of the possible coordinates for the midpoint of segment QR.
In conclusion, the possible coordinates for the midpoint of segment PQ are (6, 0) and (-2, 0), the possible coordinates for the midpoint of segment PR are (8, 0) and (-4, 0), and the midpoint of segment QR cannot be determined without specific information about the coordinates of Q and R.