If $(1{,}2)$, $(4,y)$, $(x,6)$, and $(3{,}5)$ are the vertices of a parallelogram taken in order, find $x$ and $y$.

Step 1: Understand Parallelogram Properties

A key property of a parallelogram is that its diagonals bisect each other. This means that the midpoint of one diagonal is the same as the midpoint of the other diagonal. We will use the midpoint formula to set up equations for $x$ and $y$.

Step 2: Calculate Midpoint of Diagonal AC

Let the given vertices be $A(1{,}2)$, $B(4,y)$, $C(x,6)$, and $D(3{,}5)$. The diagonals are AC and BD. First, we calculate the midpoint of diagonal AC using the coordinates of A and C.

Step 3: Calculate Midpoint of Diagonal BD

Next, we calculate the midpoint of the other diagonal, BD, using the coordinates of B and D. This will give us an expression involving $y$.

Step 4: Equate Midpoints and Solve for x and y

Since the midpoints of the diagonals are the same, we can equate their respective x-coordinates and y-coordinates. This gives us two simple equations to solve for $x$ and $y$.

More from this chapter

• Q7. If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b. [2 Marks]
• Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6).
• Find a point on the y-axis that is at the same distance from the points (-5, 2) and (9, -2).
• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
• 1. Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (-a, -b)
• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
• Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.

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