Question

35 Points!!

Verify the parallelogram ABCD with vertices A(-5, -1), B(-9, 6), C(-1, 5), and D(3, -2) is a rhombus by showing that it is a parallelogram with perpendicular diagonals.

Answer 1

- Equal midpoints of AC and BC.

- The product of the slopes of the diagonals AC and DB is -1.

Explanation:

1. Plot the given points, as you can see in the graph attached.

2. Calculate the midpoint of AC and DB:

`M_(AC)=((-5+(-1))/(2),(-1+5)/(2))=(-3,2)\\M_(DB)=((-9+3)/(2),(6+(-2))/(2))=(-3,2)`

Therefore, the midpoint of AC and DB are equal.

3. Calculte the slope of the diagonals AC and DB:

`m_(AC)=(5-(-1))/(-1-(-5))=(3)/(2)\\m_(DB)=(-2-6))/(3-(-9))=-(2)/(3)`

4. Multiply the slopes of the diagonals:

`((3)/(2))(-(2)/(3))=-1` (AC and DB are perpendicular)