two vertices of a parallelogram are a(2, 3) and b(8, 11), and the intersection of the diagonals is x(7, 6)
a. find the coordinates of the two other vertices. make sure to show all your work in finding the answer
b. explain what you had to do to find the answer. basically walking through how and why you did these things
If the usual ordering is used, the A and B are ends of one side.
The diagonals bisect each other, so (7,6) is the midpoint of AC and BD.
So, to find (x,y) of C, you have
AC = X + (X-A)
(x,y) = (7,6) + (7-2,6-3) = (7,6)+(5,3) = (12,9)
Do the same to find D.
the diagonals of a parallelogram bisect each other
so the intersection is the midpoint between a given vertex and its opposite
a. To find the coordinates of the two other vertices of the parallelogram, we can use the fact that the diagonals of a parallelogram bisect each other. In other words, the midpoint of one diagonal is the same as the midpoint of the other diagonal.
Let's call the two vertices we need to find C and D. We already know the coordinates of point X, which is the intersection of the diagonals. So, we can find the midpoint of the diagonal that connects A and B by averaging the x-coordinates and the y-coordinates separately.
Midpoint of AB = [(x-coordinate of A + x-coordinate of B)/2, (y-coordinate of A + y-coordinate of B)/2]
= [(2 + 8)/2, (3 + 11)/2]
= [10/2, 14/2]
= [5, 7]
Now, since the diagonals of a parallelogram bisect each other, the coordinates of point X (7, 6) should also be the midpoint of the diagonal that connects C and D.
Midpoint of CD = [(x-coordinate of C + x-coordinate of D)/2, (y-coordinate of C + y-coordinate of D)/2]
We can set up the following equation:
[(x-coordinate of C + x-coordinate of D)/2, (y-coordinate of C + y-coordinate of D)/2] = (7, 6)
Using the values from the midpoint calculation, we can substitute into the equation:
[(x-coordinate of C + x-coordinate of D)/2, (y-coordinate of C + y-coordinate of D)/2] = (7, 6)
[(x-coordinate of C + x-coordinate of D)/2, (y-coordinate of C + y-coordinate of D)/2] = (7, 6)
[(x-coordinate of C + x-coordinate of D)/2, (y-coordinate of C + y-coordinate of D)/2] = (7, 6)
Cross-multiplying, we get:
(x-coordinate of C + x-coordinate of D) = 2 * 7
(y-coordinate of C + y-coordinate of D) = 2 * 6
Simplifying further:
x-coordinate of C + x-coordinate of D = 14
y-coordinate of C + y-coordinate of D = 12
We now have two equations and two unknowns, which means we can solve for the coordinates of C and D.
b. To find the coordinates of the other two vertices, we used the fact that the diagonals of a parallelogram bisect each other. This means that the midpoint of one diagonal is the same as the midpoint of the other diagonal. By finding the midpoint of the diagonal that connects A and B, we obtained the coordinates (5, 7). Since the diagonals bisect each other, the coordinates of the intersection point X (7, 6) should also be the midpoint of the diagonal that connects C and D. Setting up an equation using the midpoint, we were able to solve for the coordinates of C and D by equating them to the double of the coordinates of X.
To find the coordinates of the two other vertices of the parallelogram, we can use the midpoint formula and the given information.
a. Let's assume the other vertices as C and D.
Step 1: Find the midpoint of the diagonal AC using the formula:
Midpoint of AC = ((x1 + x2) / 2, (y1 + y2) / 2)
Given:
Point A = (2, 3)
Point X = (7, 6)
Midpoint of AC = ((2 + 7) / 2, (3 + 6) / 2)
= (9/2, 9/2)
= (4.5, 4.5)
Step 2: Find the coordinates of point C by doubling the x-coordinate of point X and subtracting the x-coordinate of point B.
Coordinate of C = (2 * x-coordinate of X - x-coordinate of B, 2 * y-coordinate of X - y-coordinate of B)
Given:
Point X = (7, 6)
Point B = (8, 11)
Coordinate of C = (2 * 7 - 8, 2 * 6 - 11)
= (14 - 8, 12 - 11)
= (6, 1)
Step 3: Find the coordinates of point D by doubling the x-coordinate of point X and subtracting the x-coordinate of point A.
Coordinate of D = (2 * x-coordinate of X - x-coordinate of A, 2 * y-coordinate of X - y-coordinate of A)
Given:
Point X = (7, 6)
Point A = (2, 3)
Coordinate of D = (2 * 7 - 2, 2 * 6 - 3)
= (14 - 2, 12 - 3)
= (12, 9)
Therefore, the coordinates of the two other vertices are C(6, 1) and D(12, 9).
b. To find the coordinates of the two other vertices, we used the concept of the midpoint formula and the properties of a parallelogram.
Since we knew the coordinates of point A, point B, and the intersection point X of the diagonals, we first found the midpoint of the diagonal AC using the midpoint formula.
Next, we used the coordinates of point X, point B, and the formula for finding the coordinates of the points on a line segment in the ratio 2:1 to find the coordinates of point C.
Similarly, we used the coordinates of point X, point A, and the same formula to find the coordinates of point D.
By applying these steps, we were able to determine the coordinates of the two other vertices of the parallelogram.
To find the coordinates of the two other vertices of the parallelogram, we can utilize the fact that opposite sides of a parallelogram are parallel and equal in length.
a. Finding coordinates of vertex C:
Since the diagonals of a parallelogram bisect each other, the midpoint of the diagonal AC is the same as the midpoint of the other diagonal BD. Let's calculate the midpoint:
Midpoint formula for coordinates:
M = ((x1 + x2)/2, (y1 + y2)/2)
Given:
A(2, 3) and X(7, 6)
Using the midpoint formula:
Midpoint of AC = ((2 + 7)/2, (3 + 6)/2)
= (9/2, 9/2)
= (4.5, 4.5)
Now that we have the midpoint, C, we need to find the coordinates of C with respect to A. To do this, we can take the difference between the midpoint and A and add it to B.
Difference between midpoint and A = (4.5 - 2, 4.5 - 3)
= (2.5, 1.5)
Adding the difference to B:
C = B + Difference = (8 + 2.5, 11 + 1.5)
= (10.5, 12.5)
Therefore, the coordinates of vertex C are (10.5, 12.5).
b. Explanation:
To find vertex C, we utilized the property that diagonals of a parallelogram bisect each other. We started by finding the midpoint of diagonal AC, which is also the midpoint of diagonal BD. Then, we found the difference between the midpoint and vertex A, and added this difference to vertex B to get vertex C. This method allows us to find the coordinates of the other vertices based on the given information about the diagonals and vertices.