To find an expression for \( PM \), we can use the distance formula. The distance between point \( P(a,b) \) and the midpoint \( M(c,d) \) is given by:
\[ PM = \sqrt{(c - a)^2 + (d - b)^2} \]
Next, since \( M \) is the midpoint of the line segment \( PQ \), we can express the coordinates of \( Q \) in terms of \( P \) and \( M \). The midpoint \( M \) is defined by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( P = (x_1, y_1) \) and \( Q = (x_2, y_2) \). In our case, \( P(a, b) \) and \( M(c, d) \):
\[ M(c,d) = \left( \frac{a + x_2}{2}, \frac{b + y_2}{2} \right) \]
This gives us a system of equations:
- \( c = \frac{a + x_2}{2} \)
- \( d = \frac{b + y_2}{2} \)
From these equations, we can solve for \( x_2 \) and \( y_2 \) (the coordinates of point \( Q \)):
From equation 1: \[ c = \frac{a + x_2}{2} \implies 2c = a + x_2 \implies x_2 = 2c - a \]
From equation 2: \[ d = \frac{b + y_2}{2} \implies 2d = b + y_2 \implies y_2 = 2d - b \]
Now we have the coordinates of point \( Q \):
- \( Q = (2c - a, 2d - b) \)
In summary:
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The expression for \( PM \) is given by: \[ PM = \sqrt{(c - a)^2 + (d - b)^2} \]
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The coordinates of point \( Q \) are: \[ Q = (2c - a, 2d - b) \]