To show that the points $A(-7, -3), B(5, 10), C(15, 8), D(3, -5)$ form the vertices of a parallelogram, we will use the property that the diagonals of a parallelogram bisect each other. This means the midpoints of the diagonals should coincide.

Step 1: Find the midpoint of diagonal AC.

• Coordinates of $A = (-7, -3)$ and $C = (15, 8)$.
• The formula for the midpoint $M$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
• For diagonal AC: $M_{AC} = \left( \frac{-7 + 15}{2}, \frac{-3 + 8}{2} \right) = \left( \frac{8}{2}, \frac{5}{2} \right) = (4, 2.5)$

Step 2: Find the midpoint of diagonal BD.

• Coordinates of $B = (5, 10)$ and $D = (3, -5)$.
• Using the midpoint formula again: $M_{BD} = \left( \frac{5 + 3}{2}, \frac{10 + (-5)}{2} \right) = \left( \frac{8}{2}, \frac{5}{2} \right) = (4, 2.5)$

Step 3: Conclusion.

Since the midpoints of diagonals AC and BD are the same, the diagonals bisect each other. Therefore, the points $A(-7, -3), B(5, 10), C(15, 8), D(3, -5)$ form the vertices of a parallelogram.

Would you like further explanation, or do you have any other questions?

Related Questions:

1. How can we check if a quadrilateral is a parallelogram using vector methods?
2. What is the formula to calculate the area of a parallelogram using its vertices?
3. Can you find the lengths of the sides of the parallelogram formed by these points?
4. How can we determine if a given quadrilateral is a rectangle or square based on coordinates?
5. How can we check if the diagonals of a parallelogram are perpendicular?

Tip:

In coordinate geometry, finding midpoints and using diagonal properties are often quick methods for proving shapes like parallelograms.