ABCD is a square . if the coordinates of three of its vertices are A(-1,2a), B(a,2a), C(a,0), find the coordinates of D.
BC is obviously 2a (it's a vertical line)
then AB must be 2a (notice AB is horizontal)
so BC = AC because it is a square
2a = a+1
a=1
Now you can find the coordinates of A, B, and C, it should be easy for you to find D.
D(-1,0)
To find the coordinates of point D, we need to consider that ABCD is a square and the given coordinates of points A, B, and C.
We already have the coordinates of A, B, and C:
A(-1, 2a) where a=1
B(a, 2a) where a=1
C(a, 0) where a=1
Since ABCD is a square, we know that the length of AB is equal to the length of BC, and that AB is horizontal and BC is vertical.
The length AB is given by the difference between the x-coordinates of A and B:
AB = (a - (-1)) = a + 1 = 1 + 1 = 2
The length BC is given by the difference between the y-coordinates of B and C:
BC = (2a - 0) = 2(1) = 2
Since AB = BC, we can conclude that AB = BC = 2.
Since AB is a horizontal line, the x-coordinate of D will be a + 2.
Since BC is a vertical line, the y-coordinate of D will be the same as the y-coordinate of C, which is 0.
Therefore, the coordinates of D are:
D(a + 2, 0)
Substituting the value of a, we get:
D(1 + 2, 0) = D(3, 0)
The coordinates of D are (3, 0).
To find the coordinates of point D, we will use the fact that D is the fourth vertex of the square ABCD.
Since AB is horizontal and its length is 2a, we know that the x-coordinate of point A is -1+a. Given that the y-coordinate of A is 2a, we have the coordinates (x,y) of point A as (-1+a, 2a).
Similarly, point B has coordinates (x,y) where x=a and y=2a.
Point C has coordinates (x,y) where x=a and y=0.
Now that we have the coordinates of points A, B, and C, we can find the coordinates of point D.
Since AB is horizontal, DC must also be horizontal and equal in length to AB, so the y-coordinate of D will be same as that of C, which is 0.
To find the x-coordinate of D, we consider that AD must be vertical, and its length is 2a, just like BC. So, the x-coordinate of D will be same as that of B, which is a.
Therefore, the coordinates of point D are (a,0).
In this case, since a=1 (as determined by equating the lengths of BC and AB), the coordinates of point D are (1,0).