Finding the Intersection Point of Parallelogram Diagonals

In a parallelogram, the diagonals bisect each other. This means that the point of intersection of the two diagonals is the midpoint of each diagonal.

We are given the vertices of the parallelogram ABCD in order as A (1, 2), B (4, y), C (x, 6), and D (3, 5).

The diagonals are AC and BD.

Using the Midpoint Formula

The midpoint formula for a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\).

Calculating the Midpoint of Diagonal AC

The endpoints of diagonal AC are A (1, 2) and C (x, 6). Using the midpoint formula:

Midpoint of AC \( = \left(\frac{1+x}{2}, \frac{2+6}{2}\right) = \left(\frac{1+x}{2}, \frac{8}{2}\right) = \left(\frac{1+x}{2}, 4\right) \)

Let the point of intersection of the diagonals be P. So, P has coordinates \(\left(\frac{1+x}{2}, 4\right)\). This tells us that the y-coordinate of the intersection point is 4.

Calculating the Midpoint of Diagonal BD

The endpoints of diagonal BD are B (4, y) and D (3, 5). Using the midpoint formula:

Midpoint of BD \( = \left(\frac{4+3}{2}, \frac{y+5}{2}\right) = \left(\frac{7}{2}, \frac{y+5}{2}\right) \)

Since the point of intersection P is also the midpoint of BD, P has coordinates \(\left(\frac{7}{2}, \frac{y+5}{2}\right)\). This tells us that the x-coordinate of the intersection point is \(\frac{7}{2}\).

Determining the Point of Intersection

Since the point of intersection is the same point (P), its coordinates must be equal regardless of which diagonal's midpoint formula is used. From the midpoint of AC, we found the y-coordinate is 4. From the midpoint of BD, we found the x-coordinate is \(\frac{7}{2}\).

Thus, the point of intersection of the diagonals is \(\left(\frac{7}{2}, 4\right)\).

We can also verify this by equating the corresponding coordinates to find the values of x and y:

• Equating x-coordinates: \(\frac{1+x}{2} = \frac{7}{2}\)
• Multiplying both sides by 2: \(1+x = 7\)
• Subtracting 1 from both sides: \(x = 6\)

• Equating y-coordinates: \(4 = \frac{y+5}{2}\)
• Multiplying both sides by 2: \(8 = y+5\)
• Subtracting 5 from both sides: \(y = 3\)

So, the vertices are A(1, 2), B(4, 3), C(6, 6), and D(3, 5). Using these values, the midpoint of AC is \(\left(\frac{1+6}{2}, \frac{2+6}{2}\right) = \left(\frac{7}{2}, 4\right)\) and the midpoint of BD is \(\left(\frac{4+3}{2}, \frac{3+5}{2}\right) = \left(\frac{7}{2}, 4\right)\). Both midpoints are indeed \(\left(\frac{7}{2}, 4\right)\), confirming our result.

The point of intersection of the diagonals is \(\left(\frac{7}{2}, 4\right)\).

Conclusion

The point of intersection of the diagonals of the parallelogram with vertices A (1, 2), B (4, y), C (x, 6) and D (3, 5) taken in order is \(\left(\frac{7}{2}, 4\right)\).

Revision Table: Parallelogram Diagonals

Additional Information: Coordinate Geometry Concepts

Coordinate geometry helps us study geometric figures using a coordinate system. Key concepts include:

• Points and Coordinates: Locating points in a plane using ordered pairs (x, y).
• Distance Formula: Calculating the distance between two points.
• Midpoint Formula: Finding the coordinates of the midpoint of a line segment.
• Section Formula: Finding the coordinates of a point dividing a line segment in a given ratio.
• Slope of a Line: Measuring the steepness and direction of a line.
• Equations of Lines and Geometric Shapes: Representing lines, circles, parabolas, etc., using algebraic equations.

Understanding these concepts is fundamental for solving problems involving geometric figures in the coordinate plane, such as finding properties of parallelograms, triangles, and other polygons.