You are designing a rectangular playground. On your scale drawing, the vertices of the rectangle are (6, 3), (6, 5), and (8, 3). What are the coordinates of the fourth vertex?
(8,5)
The best way to find this is to draw the other 3 points on a graph and it becomes quite clear that you need (8,5)
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i need help whit all answers
To find the coordinates of the fourth vertex of the rectangular playground, we can use the properties of rectangles.
A rectangle has opposite sides that are equal in length and parallel to each other. In this case, since we are given three vertices of the rectangle, we can determine the length of its sides.
Let's start by finding the length of one side of the rectangle. We have two vertices with the y-coordinate of 3: (6, 3) and (8, 3). The difference between the x-coordinates of these two points will give us the length of one side of the rectangle.
Length of the rectangle = (x-coordinate of the second point) - (x-coordinate of the first point) = 8 - 6 = 2
Now, we need to find the length of the other side. We have two vertices with the x-coordinate of 6: (6, 3) and (6, 5). The difference between the y-coordinates of these two points will give us the length of the other side.
Width of the rectangle = (y-coordinate of the second point) - (y-coordinate of the first point) = 5 - 3 = 2
Since we have opposite sides of equal length, we can conclude that the rectangle is a square. In a square, all four vertices have the same distance from a center point, which means they are equidistant.
To find the fourth vertex, we can use the given vertices and calculate the coordinates based on the distance.
The given vertices are: (6, 3), (6, 5), and (8, 3).
Since the rectangle is a square, the fourth vertex will have the same distance from the center point as the given vertices.
Center point coordinates = (Average of x-coordinates, average of y-coordinates)
= ((6 + 6 + 8)/3, (3 + 5 + 3)/3)
= (20/3, 11/3)
Now, we need to find the distance between the center point and any one of the given vertices.
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's take the distance from the center point to (6, 3):
Distance = sqrt((6 - 20/3)^2 + (3 - 11/3)^2)
= sqrt((18/3 - 20/3)^2 + (9/3 - 11/3)^2)
= sqrt((-2/3)^2 + (-2/3)^2)
= sqrt((4/9) + (4/9))
= sqrt(8/9)
= sqrt(8)/3
Now, we need to find the coordinates of the fourth vertex based on this distance.
The fourth vertex will be at the same distance from the center point but in the opposite direction. Since the center point is at (20/3, 11/3), the x-coordinate of the fourth vertex will be (20/3 - sqrt(8)/3) and the y-coordinate will be (11/3 - sqrt(8)/3).
Therefore, the coordinates of the fourth vertex are:
(x, y) = (20/3 - sqrt(8)/3, 11/3 - sqrt(8)/3)
Simplifying this, we get:
(x, y) = ((20 - sqrt(8))/3, (11 - sqrt(8))/3)
So, the coordinates of the fourth vertex are ((20 - sqrt(8))/3, (11 - sqrt(8))/3).