Suppose the coordinate of P is 2, PQ = 8, and PR = 12. What are the possible coordinates of the midpoint of the given segment?
PQ = 8, PR = 12.
P(2,0), M(X,0), Q(X1,0),R(X2,0).
X2 - 2 = 12,
X2 = 14.
X2 - 2 = 2(X-2),
12 = 2X - 4,
2X = 16,
X = 8.
M(8,0) = Coordinates of mid-point.
To find the possible coordinates of the midpoint of the segment PQ, we need to determine the average of the coordinates of P and Q.
The coordinates of P are given as (2, ?). We don't know the y-coordinate of P, so we'll leave it as a variable, let's call it Py.
The coordinates of Q are not given explicitly, but we know that the length of PQ is 8. So, we can determine the x-coordinate of Q by adding 8 to the x-coordinate of P, which is 2. Therefore, the x-coordinate of Q is 2 + 8 = 10.
Now, we can find the average of the x-coordinate and y-coordinate separately, as follows:
Average x-coordinate = (Px + Qx) / 2 = (2 + 10) / 2 = 12 / 2 = 6
Average y-coordinate = (Py + Py) / 2 = 2Py / 2 = Py
Therefore, the possible coordinates of the midpoint of the segment PQ are (6, Py).
Please note that the possible y-coordinate can vary depending on the specific location of point P.
To find the midpoint of segment PQ, we can use the formula:
Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Here, point P is given as (2, _) and the length of PQ is 8 units. Therefore, we need to find the coordinates of point Q.
Since point Q is on the same line as point P, the x-coordinate of point Q will be the same as that of point P, which is 2. We can use this information to find the possible y-coordinates of point Q.
Using the Pythagorean theorem, we know that:
PQ² = (x₂ - x₁)² + (y₂ - y₁)²
Substituting the given values:
8² = (2 - 2)² + (y₂ - y₁)²
64 = (y₂ - y₁)²
Since the distance between P and Q is 8, the respective y-coordinates of the two points will differ by either +8 or -8.
Now, let's consider the other point PR. Given that point P has coordinates (2, _) and the length of PR is 12 units, we need to find the coordinates of point R.
Since point R is on the same line as point P, the x-coordinate of point R will be the same as that of point P, which is 2. We can use this information to find the possible y-coordinates of point R.
Using the Pythagorean theorem, we know that:
PR² = (x₂ - x₁)² + (y₂ - y₁)²
Substituting the given values:
12² = (2 - 2)² + (y₂ - y₁)²
144 = (y₂ - y₁)²
Similar to PQ, the distance between P and R is 12, meaning the respective y-coordinates of the two points will differ by either +12 or -12.
Therefore, the possible coordinates of the midpoint of the given segment are:
(2, y₁ + 8)
(2, y₁ - 8)
(2, y₁ + 12)
(2, y₁ - 12)
where y₁ is the y-coordinate of point P.