Answer :
To prove that quadrilateral ABCD, with vertices A(0, 4), B(3, 8), C(8, 3), and D(5, -1), is a parallelogram but not a rectangle, we need to establish two conditions: that opposite sides are both parallel and equal in length (proving it's a parallelogram), and that adjacent sides are not perpendicular (proving it's not a rectangle).
### Step 1: Calculate the Slopes of the Sides
First, let's calculate the slopes of the sides to check if opposite sides are parallel.
Slope of AB:
Slope = [tex]\( \frac{(y_2 - y_1)}{(x_2 - x_1)} \)[/tex]
[tex]\[ \text{Slope}_{AB} = \frac{8 - 4}{3 - 0} = \frac{4}{3} = 1.3333 \][/tex]
Slope of CD:
[tex]\[ \text{Slope}_{CD} = \frac{3 - (-1)}{8 - 5} = \frac{4}{3} = 1.3333 \][/tex]
Slope of BC:
[tex]\[ \text{Slope}_{BC} = \frac{3 - 8}{8 - 3} = \frac{-5}{5} = -1 \][/tex]
Slope of DA:
[tex]\[ \text{Slope}_{DA} = \frac{4 - (-1)}{0 - 5} = \frac{5}{-5} = -1 \][/tex]
Since [tex]\( \text{Slope}_{AB} = \text{Slope}_{CD} = 1.3333 \)[/tex] and [tex]\( \text{Slope}_{BC} = \text{Slope}_{DA} = -1 \)[/tex], the opposite sides are parallel.
### Step 2: Calculate the Lengths of the Sides
Next, we'll calculate the lengths of the sides to check if opposite sides are equal.
Length of AB:
Distance = [tex]\( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]
[tex]\[ \text{Length}_{AB} = \sqrt{(3 - 0)^2 + (8 - 4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \][/tex]
Length of CD:
[tex]\[ \text{Length}_{CD} = \sqrt{(8 - 5)^2 + (3 - (-1))^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \][/tex]
Length of BC:
[tex]\[ \text{Length}_{BC} = \sqrt{(8 - 3)^2 + (3 - 8)^2} = \sqrt{25 + 25} = \sqrt{50} = 7.0711 \][/tex]
Length of DA:
[tex]\[ \text{Length}_{DA} = \sqrt{(0 - 5)^2 + (4 - (-1))^2} = \sqrt{25 + 25} = \sqrt{50} = 7.0711 \][/tex]
Since [tex]\( \text{Length}_{AB} = \text{Length}_{CD} = 5 \)[/tex] and [tex]\( \text{Length}_{BC} = \text{Length}_{DA} = 7.0711 \)[/tex], the opposite sides are equal in length.
### Step 3: Check for Perpendicular Slopes (Right Angles)
To determine if ABCD is a rectangle, we need to check if any adjacent sides are perpendicular, which would make their slopes the negative reciprocal of each other.
Product of Slopes for Adjacent Sides AB and BC:
[tex]\[ \text{Slope}_{AB} \times \text{Slope}_{BC} = 1.3333 \times -1 = -1.3333 \][/tex]
Product of Slopes for Adjacent Sides CD and DA:
[tex]\[ \text{Slope}_{CD} \times \text{Slope}_{DA} = 1.3333 \times -1 = -1.3333 \][/tex]
The products are not equal to -1, hence, AB is not perpendicular to BC and CD is not perpendicular to DA. As a result, the quadrilateral does not have right angles.
### Conclusion
Since opposite sides are both parallel and equal in length, quadrilateral ABCD is a parallelogram. However, because none of the adjacent sides form a right angle, ABCD is not a rectangle.
### Step 1: Calculate the Slopes of the Sides
First, let's calculate the slopes of the sides to check if opposite sides are parallel.
Slope of AB:
Slope = [tex]\( \frac{(y_2 - y_1)}{(x_2 - x_1)} \)[/tex]
[tex]\[ \text{Slope}_{AB} = \frac{8 - 4}{3 - 0} = \frac{4}{3} = 1.3333 \][/tex]
Slope of CD:
[tex]\[ \text{Slope}_{CD} = \frac{3 - (-1)}{8 - 5} = \frac{4}{3} = 1.3333 \][/tex]
Slope of BC:
[tex]\[ \text{Slope}_{BC} = \frac{3 - 8}{8 - 3} = \frac{-5}{5} = -1 \][/tex]
Slope of DA:
[tex]\[ \text{Slope}_{DA} = \frac{4 - (-1)}{0 - 5} = \frac{5}{-5} = -1 \][/tex]
Since [tex]\( \text{Slope}_{AB} = \text{Slope}_{CD} = 1.3333 \)[/tex] and [tex]\( \text{Slope}_{BC} = \text{Slope}_{DA} = -1 \)[/tex], the opposite sides are parallel.
### Step 2: Calculate the Lengths of the Sides
Next, we'll calculate the lengths of the sides to check if opposite sides are equal.
Length of AB:
Distance = [tex]\( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]
[tex]\[ \text{Length}_{AB} = \sqrt{(3 - 0)^2 + (8 - 4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \][/tex]
Length of CD:
[tex]\[ \text{Length}_{CD} = \sqrt{(8 - 5)^2 + (3 - (-1))^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \][/tex]
Length of BC:
[tex]\[ \text{Length}_{BC} = \sqrt{(8 - 3)^2 + (3 - 8)^2} = \sqrt{25 + 25} = \sqrt{50} = 7.0711 \][/tex]
Length of DA:
[tex]\[ \text{Length}_{DA} = \sqrt{(0 - 5)^2 + (4 - (-1))^2} = \sqrt{25 + 25} = \sqrt{50} = 7.0711 \][/tex]
Since [tex]\( \text{Length}_{AB} = \text{Length}_{CD} = 5 \)[/tex] and [tex]\( \text{Length}_{BC} = \text{Length}_{DA} = 7.0711 \)[/tex], the opposite sides are equal in length.
### Step 3: Check for Perpendicular Slopes (Right Angles)
To determine if ABCD is a rectangle, we need to check if any adjacent sides are perpendicular, which would make their slopes the negative reciprocal of each other.
Product of Slopes for Adjacent Sides AB and BC:
[tex]\[ \text{Slope}_{AB} \times \text{Slope}_{BC} = 1.3333 \times -1 = -1.3333 \][/tex]
Product of Slopes for Adjacent Sides CD and DA:
[tex]\[ \text{Slope}_{CD} \times \text{Slope}_{DA} = 1.3333 \times -1 = -1.3333 \][/tex]
The products are not equal to -1, hence, AB is not perpendicular to BC and CD is not perpendicular to DA. As a result, the quadrilateral does not have right angles.
### Conclusion
Since opposite sides are both parallel and equal in length, quadrilateral ABCD is a parallelogram. However, because none of the adjacent sides form a right angle, ABCD is not a rectangle.
