Three consecutive vertices of a parallelogram are points (2,4) (0,0) and (6,0). What is the fourth vertice point?
First: >1 vertices; 1 vertex
Make a diagram. It should be easy to see where the last point goes.
You will have two sets of parallel sides. The base is the line from (0,0) to (6,0).
The third point is over 2 and up 4 from (0,0) so the last point will be over 2 and up 4 from (6,0) at (8,4)
To find the fourth vertex of a parallelogram, we need to understand the properties of a parallelogram. One property of a parallelogram is that opposite sides are parallel, which means they have the same slope. Another property is that opposite sides are equal in length.
Given three consecutive vertices of the parallelogram: (2,4), (0,0), and (6,0), we can use this information to find the slope of one of the sides and the length of one of the sides.
Let's start by finding the slope of one of the sides. We can use the formula for slope, which is (change in y) / (change in x).
Let's take the vertices (2,4) and (0,0) as our first side. The change in y is 4 - 0 = 4, and the change in x is 2 - 0 = 2.
Therefore, the slope of this side is 4/2 = 2.
Now that we have the slope of one side, we need to find the length of that side. We can use the distance formula to calculate the length between two points.
Let's take the vertices (2,4) and (0,0). The distance formula is √[(x2 - x1)^2 + (y2 - y1)^2].
Using the distance formula, we have:
Distance = √[(0 - 2)^2 + (0 - 4)^2]
= √[(-2)^2 + (-4)^2]
= √[4 + 16]
= √20
= 2√5
Therefore, the length of our first side is 2√5.
Now we can use this information to find the fourth vertex. Since opposite sides of a parallelogram are equal in length, the length of the fourth side must also be 2√5.
Let's assume the fourth vertex is (x, y).
We know that one side of the parallelogram is formed by the vertices (2,4) and (0,0), and its slope is 2. Therefore, the equation of this side is:
y - 4 = 2(x - 2)
Expanding and rearranging the equation, we get:
y - 4 = 2x - 4
y = 2x
Now, we can use the length of the fourth side to find its coordinates.
The distance between (0,0) and (x, y) is 2√5. We can use the distance formula again:
2√5 = √[(x - 0)^2 + (y - 0)^2]
4*5 = x^2 + y^2
20 = x^2 + y^2
Since we know that y = 2x, we can substitute it into the equation:
20 = x^2 + (2x)^2
20 = x^2 + 4x^2
20 = 5x^2
x^2 = 20/5
x^2 = 4
x = ±√4
x = ±2
If x = 2, then y = 2(2) = 4.
If x = -2, then y = 2(-2) = -4.
So, the coordinates of the fourth vertex can be either (2, 4) or (-2, -4).