Parallelogram Coordinate Geometry Solver
Let's solve this problem together, step-by-step!
The Problem
If the points A(1, 2), B(4, y), C(x, 6), and D(3, 5) are the vertices of a parallelogram taken in order, find the values of x and y.
1. Understanding the Terms
Vertices: These are simply the corners of a shape. In our case, we have four vertices: A, B, C, and D.
Parallelogram: A four-sided flat shape where opposite sides are parallel and equal in length. For this problem, the most important property is that its diagonals bisect each other. This means they cross at their exact midpoints.
2. What is Given?
We are given the coordinates of the four vertices in order:
- A = (1, 2)
- B = (4, y)
- C = (x, 6)
- D = (3, 5)
3. What Do We Need to Find?
Our goal is to find the numeric values for the two unknowns: x and y.
4. How Can We Solve It? The Key Concept
As we mentioned, the diagonals of a parallelogram bisect each other. This means the midpoint of diagonal AC is the exact same point as the midpoint of diagonal BD.
To find the midpoint of a line segment between two points (x1, y1) and (x2, y2), we use the Midpoint Formula:
5. Step-by-Step Solution
Step A: Find the midpoint of diagonal AC
Using points A(1, 2) and C(x, 6):
Step B: Find the midpoint of diagonal BD
Using points B(4, y) and D(3, 5):
Step C: Equate the Midpoints
We know that diagonals of a parallelogram bisect each other. So, the coordinates of the midpoint of AC must be equal to the coordinates of the midpoint of BD.
i.e.,
Now, we equate the corresponding x and y coordinates to solve for our unknowns:
6. Now, You Solve for x and y!
Use the equations from Step C to find the final answer.
Solution Walkthrough:
Solving for x:
Multiply both sides by 2: 1 + x = 7
Subtract 1 from both sides: x = 6
Solving for y:
Multiply both sides by 2: 8 = y + 5
Subtract 5 from both sides: 3 = y
So, the final answer is x = 6 and y = 3.
7. Challenge Problem #1
If the points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram, taken in order, find the value of p.
Challenge Solution Walkthrough:
Just like before, the midpoint of AC must equal the midpoint of BD.
Midpoint of AC: Using A(6, 1) and C(9, 4)
Midpoint of BD: Using B(8, 2) and D(p, 3)
Equate the x-coordinates: The y-coordinates are already equal (5/2).
Multiply both sides by 2: 15 = 8 + p
Subtract 8 from both sides: 7 = p
So, the final answer is p = 7.
8. Challenge Problem #2
The points A(2, 4), B(x, 7), C(8, 9), D(6, y) are the vertices of a parallelogram. Find x and y.
Challenge Solution Walkthrough:
Midpoint of AC = Midpoint of BD.
Midpoint of AC: Using A(2, 4) and C(8, 9)
Midpoint of BD: Using B(x, 7) and D(6, y)
Equate coordinates:
x-coords:
y-coords:
9. Challenge Problem #3
The points A(-2, -1), B(1, 0), C(x, 3), D(1, y) are the vertices of a parallelogram. Find x and y.
Challenge Solution Walkthrough:
Midpoint of AC = Midpoint of BD.
Midpoint of AC: Using A(-2, -1) and C(x, 3)
Midpoint of BD: Using B(1, 0) and D(1, y)
Equate coordinates:
x-coords:
y-coords: 1 =