For a line segment, one endpoint is (5,-7) and the midpoint is (7,3). Find the other endpoint.

Steps to Solve

1. Identify Given Values
   We know one endpoint ( A(5, -7) ) and the midpoint ( M(7, 3) ).

2. Use the Midpoint Formula
   The midpoint ( M ) of a line segment connecting points ( A(x_1, y_1) ) and ( B(x_2, y_2) ) is given by:
   $$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$
   In this case, we'll consider ( A(5, -7) ) as ( (x_1, y_1) ) and find ( B(x_2, y_2) ).

3. Set Up Equations for ( x ) and ( y )
   From the midpoint formula, we have:

• For the ( x )-coordinates:
  $$ 7 = \frac{5 + x_2}{2} $$
• For the ( y )-coordinates:
  $$ 3 = \frac{-7 + y_2}{2} $$

4. Solve for ( x_2 )
   Multiply both sides of the equation for the ( x )-coordinates by 2:
   $$ 14 = 5 + x_2 $$
   Now subtract 5 from both sides:
   $$ x_2 = 14 - 5 $$
   Thus,
   $$ x_2 = 9 $$

5. Solve for ( y_2 )
   Multiply both sides of the equation for the ( y )-coordinates by 2:
   $$ 6 = -7 + y_2 $$
   Now add 7 to both sides:
   $$ y_2 = 6 + 7 $$
   Thus,
   $$ y_2 = 13 $$

6. Conclusion
   The other endpoint ( B ) is ( (9, 13) ).