The points (1,5), (4,5), and (1,0) represent three corners of a rectangle; find the coordinates of the fourth poin
Those three coordinates should tell you that the rectangle has sides parallel to the x and y axes. This makes the problem simple. The first two points are the ends of a horizontal line at y =5, with length 3.
The first and third points are the ends of a vertical line at x=0 with length 5.
A point at (4,0) will complete the rectangle with a side at x=4 of length 5, and a side at y=0 with length 3.
If you can't follow my argument, try plotting the points on a graph.
To find the coordinates of the fourth point of the rectangle, we can use the fact that opposite sides of a rectangle are parallel and equal in length.
The given points are (1,5), (4,5), and (1,0).
Let's call the fourth point (x, y).
Since the opposite sides are equal in length, the distance between (1,5) and (4,5) should be equal to the distance between (1,5) and (x, y).
Using the distance formula, we can calculate the distance between two points:
d = √((x2 - x1)^2 + (y2 - y1)^2)
We have:
d1 = √((4 - 1)^2 + (5 - 5)^2)
d2 = √((x - 1)^2 + (y - 5)^2)
Since opposite sides of a rectangle are equal, we can set the distances equal to each other:
d1 = d2
√((4 - 1)^2 + (5 - 5)^2) = √((x - 1)^2 + (y - 5)^2)
Simplifying the equation:
√(3^2 + 0^2) = √((x - 1)^2 + (y - 5)^2)
√(9 + 0) = √((x - 1)^2 + (y - 5)^2)
√9 = √((x - 1)^2 + (y - 5)^2)
3 = √((x - 1)^2 + (y - 5)^2)
Now let's consider the distance between (1,0) and (x, y):
d3 = √((x - 1)^2 + (y - 0)^2)
Since opposite sides of a rectangle are equal, we can set the distances d2 and d3 equal to each other:
√((x - 1)^2 + (y - 5)^2) = √((x - 1)^2 + (y - 0)^2)
Simplifying the equation:
√((x - 1)^2 + (y - 5)^2) = √((x - 1)^2 + y^2)
Squaring both sides of the equation:
(x - 1)^2 + (y - 5)^2 = (x - 1)^2 + y^2
Expanding the squared terms:
x^2 - 2x + 1 + y^2 - 10y + 25 = x^2 - 2x + 1 + y^2
Canceling out like terms:
- 2x - 10y + 26 = 0
This equation represents the line that passes through the points (1,5), (4,5), and (1,0).
To find the coordinates of the fourth point, we need to find the intersection point of this line with the x-axis or y-axis.
Considering the line intersecting the x-axis, we set y = 0 and solve for x:
-2x - 10(0) + 26 = 0
-2x + 26 = 0
-2x = -26
x = 13
So, one possible coordinate of the fourth point is (13,0).
Considering the line intersecting the y-axis, we set x = 0 and solve for y:
-2(0) - 10y + 26 = 0
-10y + 26 = 0
-10y = -26
y = 2.6
So, another possible coordinate of the fourth point is (0,2.6).
Therefore, the coordinates of the fourth point of the rectangle could be either (13,0) or (0,2.6).
To find the coordinates of the fourth point of a rectangle, we can make use of the properties of rectangles.
In a rectangle, opposite sides are parallel, and the lengths of opposite sides are equal. Additionally, the diagonals of a rectangle are congruent and bisect each other.
Given the points (1,5), (4,5), and (1,0), let's plot these points on a graph to visualize the rectangle.
First, plot the point (1,5). Then, using a ruler or a straight edge, draw a line segment from this point to the point (4,5). This forms one side of the rectangle. Next, draw another line segment from the point (4,5) to the point (1,0). These two line segments should meet at a right angle, forming a rectangle.
To find the coordinates of the fourth point, you need to determine the missing x and y values. One approach is to observe the given points.
We can notice that the x-coordinate of the missing point is the same as the second point's x-coordinate, which is 4. Similarly, the y-coordinate of the missing point is the same as the third point's y-coordinate, which is 0.
Therefore, the coordinates of the fourth point are (4,0).
Alternatively, we can also compute the difference between the x-coordinates and y-coordinates of the given points. If we find that the differences are non-zero and equal, it confirms that the given points form a rectangle, and the missing point will complete it.
Let's calculate the differences:
For the x-coordinates:
4 - 1 = 3
For the y-coordinates:
0 - 5 = -5
Since the differences are non-zero and equal in magnitude (3 and -5), we can conclude that the given points indeed form a rectangle. By adding the differences to the known x and y coordinates, we get 4 for the missing x-coordinate and 0 for the missing y-coordinate, confirming that the fourth point is (4,0).