To find an expression for the distance PM, where P is the point \(P(a, b)\) and M is the midpoint \(M(c, d)\), we can use the distance formula. The distance PM is given by:
\[ PM = \sqrt{(c - a)^2 + (d - b)^2} \]
Now, we also know that \(M\) is the midpoint of line segment \(PQ\). By definition, the coordinates of the midpoint \(M\) can be expressed in terms of the coordinates of points \(P\) and \(Q\):
\[ M\left(c, d\right) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
where \(P(x_1, y_1) = (a, b)\) and \(Q(x_2, y_2)\). This gives us:
\[ c = \frac{a + x_2}{2} \quad \text{and} \quad d = \frac{b + y_2}{2} \]
From these equations, we can solve for \(x_2\) and \(y_2\) (the coordinates of point \(Q\)):
\[ x_2 = 2c - a \] \[ y_2 = 2d - b \]
Now, substituting these into the expression for \(PQ\) (the length of the line segment from P to Q), we have:
\[ PQ = \sqrt{(x_2 - a)^2 + (y_2 - b)^2} \]
Substituting the expressions we obtained for \(x_2\) and \(y_2\):
\[ PQ = \sqrt{((2c - a) - a)^2 + ((2d - b) - b)^2} \]
Simplifying this:
\[ PQ = \sqrt{(2c - 2a)^2 + (2d - 2b)^2} \]
Factoring out the 2:
\[ PQ = \sqrt{4((c - a)^2 + (d - b)^2)} = 2\sqrt{(c - a)^2 + (d - b)^2} \]
Thus, the final expressions are:
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The distance PM is given by: \[ PM = \sqrt{(c - a)^2 + (d - b)^2} \]
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The length of segment PQ is given by: \[ PQ = 2 \times PM \]