Certainly! Let's find the coordinates of point [tex]\( B \)[/tex] given that the midpoint [tex]\( M \)[/tex] of line segment [tex]\( AB \)[/tex] is at [tex]\( (2, 5) \)[/tex] and point [tex]\( A \)[/tex] is at [tex]\( (1, 7) \)[/tex].
To solve this problem, we'll use the midpoint formula. The midpoint [tex]\( M \)[/tex] of a line segment connecting points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] is given by:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Given:
- Midpoint [tex]\( M = (2, 5) \)[/tex]
- Coordinates of [tex]\( A = (1, 7) \)[/tex]
We need to find the coordinates of [tex]\( B \)[/tex], let them be [tex]\( (x, y) \)[/tex].
Using the midpoint formula, we get two equations:
[tex]\[
\frac{x_1 + x_2}{2} = 2 \quad \text{and} \quad \frac{y_1 + y_2}{2} = 5
\][/tex]
Plugging in the coordinates of [tex]\( A \)[/tex] (where [tex]\( x_1 = 1 \)[/tex] and [tex]\( y_1 = 7 \)[/tex]), we have:
[tex]\[
\frac{1 + x}{2} = 2
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
1 + x = 4 \\
x = 3
\][/tex]
Next, we solve for [tex]\( y \)[/tex]:
[tex]\[
\frac{7 + y}{2} = 5
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
7 + y = 10 \\
y = 3
\][/tex]
Therefore, the coordinates of point [tex]\( B \)[/tex] are [tex]\( (3, 3) \)[/tex].
So, point [tex]\( B \)[/tex] has the coordinates [tex]\( (3, 3) \)[/tex].
