Show that the following points taken in order form the vertices of a parallelogram.  A (–7, –3), B(5,10), C(15,8), D(3, –5)

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.

Learn how to show that the points A(-7, -3), B(5,10), C(15,8), D(3, -5) form a parallelogram using the midpoint formula and the property that diagonals bisect each other. This step-by-step solution is perfect for high school students in Grades 9-11 learning coordinate geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation