Question

If \(\mathrm{P}(1,2), \mathrm{Q}(4,6), \mathrm{R}(5,7)\) and \(\mathrm{S}(\mathrm{a}, \mathrm{b})\) are the vertices of a parallelogram PQRS then (a) \(a=2, b=4\) (b) \(a=3, b=4\) (c) \(\mathrm{a}=2, \mathrm{~b}=3\) (d) \(a=2, b=5\)

Short answer

The coordinates of vertex S are (a = 2, b = 4) which makes option (a) the correct answer.

Step-by-step solution

Compute slopes for given vertices

First, we need to compute the slopes of the opposite sides of the parallelogram. We will compute the slope of PQ and the slope of R(a, b). The slope of PQ is given by: \(m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6-2}{4-1} = \frac{4}{3}\) Now let's compute the slope of R(a, b), using the given point R(5,7) and a generic point S with coordinates (a, b): \(m_{RS} = \frac{b-7}{a-5}\)

Compare the slopes

Since the opposite sides of a parallelogram have equal slopes, we will set the slope of PQ equal to the slope of R(a, b) and find the relationship between a and b: \(\frac{4}{3} = \frac{b-7}{a-5}\)

Test the given options

Now we will test each given option for the coordinates of S(a, b) against the relationship found in Step 2: (a) a=2, b=4: \(\frac{4}{3} = \frac{4-7}{2-5}\) → \(\frac{4}{3} = \frac{-3}{-3}\) which is true. (b) a=3, b=4: \(\frac{4}{3} = \frac{4-7}{3-5}\) → \(\frac{4}{3} = \frac{-3}{-2}\) which is false. (c) a=2, b=3: \(\frac{4}{3} = \frac{3-7}{2-5}\) → \(\frac{4}{3} = \frac{-4}{-3}\) which is false. (d) a=2, b=5: \(\frac{4}{3} = \frac{5-7}{2-5}\) → \(\frac{4}{3} = \frac{-2}{-3}\) which is false. From the above analysis, only option (a) holds true.

Conclusion

The coordinates of vertex S are (a=2, b=4). Option (a) is the correct answer.

Key concepts

Coordinate Geometry

Coordinate geometry is an area of mathematics that helps us study geometric figures like lines and polygons using the coordinate plane. In a coordinate plane, every point is represented by a pair of numbers, or coordinates, which define its position in space.

These numbers are expressed as \((x, y)\), where:

• \(x\) is the horizontal position, measured along the X-axis
• \(y\) is the vertical position, measured along the Y-axis

By utilizing these coordinates, we can easily perform various calculations, like finding distances, midpoints, and slopes. Coordinate geometry simplifies the exploration of geometric shapes by transforming them into algebraic equations. For example, the vertices of a shape like a parallelogram can be expressed through these coordinates, allowing us to use mathematical operations and equations to study its properties and solve problems.

Slope of a Line

The slope of a line is a fundamental concept in coordinate geometry. It measures the steepness and direction of a line. Slope is calculated as the ratio of the vertical change to the horizontal change between two points on a line. This is often referred to as "rise over run."

For any two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is calculated as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• Positive slope: line rises from left to right
• Negative slope: line falls from left to right
• Zero slope: horizontal line
• Undefined slope: vertical line

In the context of a parallelogram, opposite sides must have equal slopes, as they are parallel. This property is crucial when solving problems related to the vertices of a parallelogram, as it allows us to set up equations for finding unknown coordinates.

Equation Solving

Equation solving is a key mathematical skill, used to find unknown values that make an equation true. In the parallelogram vertices problem, we equate the slopes of opposite sides to find the unknown coordinates of a point.

To solve the equation \(\frac{4}{3} = \frac{b-7}{a-5}\), we follow these steps:

• Substitute the given values of \(a\) and \(b\) from each option.
• Simplify both sides of the equation to verify if they are equal.

This method allows us to systematically test multiple possibilities to find the correct coordinates that satisfy the equation. It's important to simplify equations accurately to ensure correct solutions.

Vertices of Parallelogram

The vertices of a parallelogram are crucial to understanding its geometric properties. A parallelogram is a four-sided figure with opposite sides that are both parallel and equal in length. The points \((x_1, y_1), (x_2, y_2), (x_3, y_3), \text{and} (x_4, y_4)\) denote these vertices.

Key characteristics of parallelograms include:

• Opposite sides are parallel and equal.
• Opposite angles are equal.
• Diagonals bisect each other.

In problems involving the vertices of a parallelogram, the goal often involves finding missing coordinates using known points. Calculating slopes, as shown in our problem, can help verify that opposite sides are truly parallel, confirming the shape's identity as a parallelogram.