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Part 1
Suppose PQ overbar has​ P(a,b) and midpoint​ M(c,d). What is an expression for​ PM? Use the expression for PM to find an expression for PQ.
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Part 1
PMequals

1 answer

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:

  1. \( c = \frac{a + x_2}{2} \)
  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:

  • The expression for \( PM \) is given by: \[ PM = \sqrt{(c - a)^2 + (d - b)^2} \]

  • The coordinates of point \( Q \) are: \[ Q = (2c - a, 2d - b) \]