To find the coordinates of point \(\( P \)\) given the midpoint \(\( M \)\) and point \(\( Q \)\), we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint \(\( M(x_M, y_M) \)\) between two points \(\( P(x_P, y_P) \)\) and \(\( Q(x_Q, y_Q) \)\) are given by:
\(\[
M_x = \frac{x_P + x_Q}{2}
\]\)
\(\[
M_y = \frac{y_P + y_Q}{2}
\]\)
Given:
- Midpoint \(\( M(10.05, 10.3) \)\)
- Point \(\( Q(13.4, 9.4) \)\)
We need to find the coordinates of point \(\( P(x_P, y_P) \)\).
### Step-by-Step Solution:
1. Write down the midpoint formulas:
\(\[
10.05 = \frac{x_P + 13.4}{2}
\]\)
\(\[
10.3 = \frac{y_P + 9.4}{2}
\]\)
2. Solve for \(\( x_P \)\):
Multiply both sides of the equation \(\( 10.05 = \frac{x_P + 13.4}{2} \)\) by 2 to get rid of the fraction:
\(\[
2 \times 10.05 = x_P + 13.4
\]\)
\(\[
20.1 = x_P + 13.4
\]\)
Subtract 13.4 from both sides to solve for \(\( x_P \)\):
\(\[
20.1 - 13.4 = x_P
\]\)
\(\[
x_P = 6.7
\]\)
3. Solve for \(\( y_P \)\):
Multiply both sides of the equation \(\( 10.3 = \frac{y_P + 9.4}{2} \)\) by 2:
\(\[
2 \times 10.3 = y_P + 9.4
\]\)
\(\[
20.6 = y_P + 9.4
\]\)
Subtract 9.4 from both sides to solve for \(\( y_P \)\):
\(\[
20.6 - 9.4 = y_P
\]\)
\(\[
y_P = 11.2
\]\)
Thus, the coordinates of point \(\( P \)\) are \(\( (6.7, 11.2) \)\).
Answer:
\(\( P = (6.7, 11.2) \)\)
