Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.