To find the coordinates of the other endpoint [tex]\(Q\)[/tex], given that the midpoint [tex]\(M\)[/tex] and one endpoint [tex]\(P\)[/tex] are known, we can use the midpoint formula.
The midpoint [tex]\(M(x_m, y_m)\)[/tex] of a segment with endpoints [tex]\(P(x_1, y_1)\)[/tex] and [tex]\(Q(x_2, y_2)\)[/tex] is given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Since we know [tex]\(M(3,5)\)[/tex] and [tex]\(P(2,5)\)[/tex], we can set up the equations:
[tex]\[ 3 = \frac{2 + x_2}{2} \][/tex]
[tex]\[ 5 = \frac{5 + y_2}{2} \][/tex]
Solving for [tex]\(x_2\)[/tex]:
Multiply both sides of the first equation by 2:
[tex]\[ 2 \left( 3 \right) = 2 + x_2 \][/tex]
[tex]\[ 6 = 2 + x_2 \][/tex]
Subtract 2 from both sides to isolate [tex]\(x_2\)[/tex]:
[tex]\[ 6 - 2 = x_2 \][/tex]
[tex]\[ x_2 = 4 \][/tex]
Solving for [tex]\(y_2\)[/tex]:
Multiply both sides of the second equation by 2:
[tex]\[ 2 \left( 5 \right) = 5 + y_2 \][/tex]
[tex]\[ 10 = 5 + y_2 \][/tex]
Subtract 5 from both sides to isolate [tex]\(y_2\)[/tex]:
[tex]\[ 10 - 5 = y_2 \][/tex]
[tex]\[ y_2 = 5 \][/tex]
Therefore, the coordinates of the other endpoint [tex]\(Q\)[/tex] are:
[tex]\[ Q = (4, 5) \][/tex]
The coordinates of [tex]\(Q\)[/tex] are [tex]\(4, 5\)[/tex].
