A triangle has vertices A(0,2), B(3,7), C(0,6). Given that ABCD is a parallelogram, find i)coordinates of D
Well, first draw a picture
now
from A or C go right 3 so x = 3
from B go down 4 or from A go up one
so y = 3
so
D(3,3)
To find the coordinates of point D, we need to understand the properties of a parallelogram. One property of a parallelogram is that opposite sides are equal in length and parallel to each other.
In this case, one of the sides of the parallelogram is given by the line segment AB. We can find the length and direction of this side using the coordinates of points A and B.
The length of a line segment can be calculated using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the length of AB:
d_AB = sqrt((3 - 0)^2 + (7 - 2)^2)
= sqrt(9 + 25)
= sqrt(34)
Now, we know that AD must have the same length as BC since they are opposite sides of the parallelogram. The length of BC can be calculated using the distance formula as well.
d_BC = sqrt((0 - 0)^2 + (6 - 7)^2)
= sqrt(0 + 1)
= sqrt(1)
= 1
So, AD also has a length of 1.
Now that we have the length of AD, we can determine its direction. Since AB is a diagonal of the parallelogram, AD must have the same slope as AB for the sides to be parallel.
The slope of AB can be calculated using the formula:
m_AB = (y2 - y1) / (x2 - x1)
m_AB = (7 - 2) / (3 - 0)
= 5 / 3
Therefore, the slope of AD is also 5/3.
To determine the coordinates of point D, we start from point A (0, 2) and move a distance of 1 unit in the direction of the slope. Since the slope is positive, we move to the right and up from point A.
Using the slope-intercept form of a linear equation (y = mx + b), we can find the y-coordinate of point D:
y_D = y_A + (m_AB * d_AD)
y_D = 2 + (5/3 * 1)
= 2 + 5/3
= 11/3
Now, let's determine the x-coordinate of point D using the slope:
x_D = x_A + (d_AD / sqrt(1 + m_AB^2))
x_D = 0 + (1 / sqrt(1 + (5/3)^2))
= 0 + (1 / sqrt(1 + 25/9))
= 0 + (1 / sqrt(34/9))
= 0 + (1 / (sqrt(34) / 3))
= 0 + (3 / sqrt(34))
= 3 / sqrt(34)
Therefore, the coordinates of point D are approximately D(3/√34, 11/3).