Question

Question content area top
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.
Question content area bottom
Part 1
PMequals
▼

Answer 1

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:
\[
Q = (2c - a, 2d - b)
\]