graph the quadrilateral separately nd algebraically and show if it's a square
quad abcd:a(0,a) b(a,2a) c(2a,a) d(a,0)
To graph the quadrilateral ABCD and determine if it is a square, we can follow these steps:
1. Plot the given points A(0, a), B(a, 2a), C(2a, a), and D(a, 0) on a coordinate plane.
2. Connect the plotted points to form the quadrilateral ABCD.
3. Calculate the distances between the four sides of the quadrilateral using the distance formula: √[(x2 - x1)^2 + (y2 - y1)^2].
4. Compare the lengths of the sides to determine if they are all equal.
5. Additionally, check if the quadrilateral's opposite sides are parallel by comparing their slopes.
Here's how we can perform the calculations:
1. Plot the points A(0, a), B(a, 2a), C(2a, a), and D(a, 0) on a coordinate plane as follows:
- Point A(0, a): If a specific value for 'a' is given, substitute it; otherwise, leave it as a variable.
- Point B(a, 2a)
- Point C(2a, a)
- Point D(a, 0)
2. Connect the plotted points to form the quadrilateral ABCD.
3. Calculate the lengths of the sides using the distance formula:
- Side AB: √[(x2 - x1)^2 + (y2 - y1)^2]
Substituting the coordinates of A and B:
AB = √[(a - 0)^2 + (2a - a)^2] = √[(a)^2 + (a)^2] = √[2a^2] = √2a
- Side BC: √[(x2 - x1)^2 + (y2 - y1)^2]
Substituting the coordinates of B and C:
BC = √[(2a - a)^2 + (a - 2a)^2] = √[(a)^2 + (-a)^2] = √[2a^2] = √2a
- Side CD: √[(x2 - x1)^2 + (y2 - y1)^2]
Substituting the coordinates of C and D:
CD = √[(a - 2a)^2 + (0 - a)^2] = √[(-a)^2 + (-a)^2] = √[2a^2] = √2a
- Side DA: √[(x2 - x1)^2 + (y2 - y1)^2]
Substituting the coordinates of D and A:
DA = √[(0 - a)^2 + (a - 0)^2] = √[(^a2)^2 + (a)^2] = √[2a^2] = √2a
4. Now, check if all the side lengths are equal. Since AB = BC = CD = DA = √2a, it indicates that all sides are equal.
5. To determine if it is a square, we need to verify if opposite sides are parallel by comparing their slopes:
- The slope of AB: (Change in y)/(Change in x) = (2a - a)/(a - 0) = a/a = 1
- The slope of BC: (Change in y)/(Change in x) = (a - 2a)/(2a - a) = -a/a = -1
- The slope of CD: (Change in y)/(Change in x) = (0 - a)/(a - 2a) = -a/(-a) = 1
- The slope of DA: (Change in y)/(Change in x) = (a - 0)/(0 - a) = a/(-a) = -1
Here, we observe that AB and CD have a slope of 1 while BC and DA have a slope of -1. This indicates that opposite sides are parallel.
Therefore, based on the equal side lengths and parallel opposite sides, the quadrilateral ABCD is a square.