Question

The three vertices of a parallelogram taken in a order are \((-1,0),(3,1)\) and \((2,2)\) respectively. Find the coordinates of the fourth vertex : (a) \((-1,2)\) (b) \((-2,1)\) (c) \((2,3)\) (d) \((3,-2)\)

Short answer

Answer: The coordinates of the fourth vertex are (-1, 1).

Step-by-step solution

Identify the given points

The given vertices of the parallelogram are: A \((-1,0)\), B \((3,1)\), and C \((2,2)\).

Calculate the midpoint of diagonal AC

Using the midpoint formula, we find the midpoint of diagonal AC, which we can call M: M = \(\left(\frac{-1+2}{2},\frac{0+2}{2}\right)\) M = \(\left(\frac{1}{2},1\right)\)

Apply the midpoint theorem to find the coordinates of the fourth vertex

Since M is the midpoint of both diagonals (AC and BD) in a parallelogram, we can find the coordinates of the fourth vertex D by applying the midpoint theorem in reverse: D = \(\left(2x_M - x_B, 2y_M - y_B\right)\) D = \(\left(2\cdot\frac{1}{2} - 3, 2\cdot1 - 1\right)\) D = \(\left(-1, 1\right)\) The coordinates of the fourth vertex D is \((-1, 1)\), which corresponds to option (b).

Key concepts

Midpoint Formula

The midpoint formula is a fundamental tool in coordinate geometry that allows us to find the exact center point between two points. If we have two points, A \( (x_{A}, y_{A}) \) and B \( (x_{B}, y_{B}) \), the midpoint, M \( (x_{M}, y_{M}) \), is calculated using the formula:

\[ x_{M} = \frac{x_{A} + x_{B}}{2} \]
\[ y_{M} = \frac{y_{A} + y_{B}}{2} \]
In the context of the given exercise, the midpoint formula helps us determine the center point of the diagonal AC in the parallelogram. This is a crucial step that leads to finding the position of the missing fourth vertex, as the diagonals of a parallelogram bisect each other. Understanding and applying this formula is key to solving many geometry problems that involve finding midpoint, distance, or partitioning segments.

Parallelogram Properties

A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. It has the following key properties that are especially useful in coordinate geometry:

• Opposite angles are equal.
• Consecutive angles are supplementary.
• Opposite sides are equal in length and parallel.
• The diagonals bisect each other.
• Each diagonal divides the parallelogram into two congruent triangles.

When working with parallelograms in a coordinate plane, these properties allow us to use algebraic methods to find missing information. For example, knowing that the diagonals bisect each other is essential in using the midpoint formula to find the coordinates of an unknown vertex. By invoking the congruency of triangles created by a diagonal, we can deduce the necessary equations to locate missing points. A clear grasp of these specific attributes greatly aids in solving complex geometric problems.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebraic equations with geometric figures. This field of math enables us to solve geometrical problems using a coordinate system, typically represented by the x and y-axes. Here are several core aspects of coordinate geometry:

• Analyzing the properties and equations of various geometric shapes like lines, circles, and polygons.
• Using formulas like the midpoint, distance, and slope to solve problems related to these shapes.
• Interpreting geometric figures in a Cartesian Plane to find area, perimeter, or other attributes.

In our exercise, coordinate geometry comes into play through the use of algebraic formulas to find the coordinates of vertices of the parallelogram. The understanding of how geometric figures translate into a system of coordinates, and vice versa, is fundamental for students tackling similar problems. The exercise draws upon these concepts and illustrates how they interconnect to reach a solution.