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