Problem 41 - Algebra and Trigonometry

Question

Which of the points \(P(3,1)\) and \(Q(-1,3)\) is closer to the point \(R(-1,-1) ?\)

Step-by-Step Solution

Answer

Point Q(-1,3) is closer to R(-1,-1) than P(3,1).

Step 1: Calculate the Distance from P to R

The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]We need to find the distance from point \(P(3, 1)\) to point \(R(-1, -1)\). Substitute the given points into the formula:\[ d_{PR} = \sqrt{((-1) - 3)^2 + ((-1) - 1)^2} = \sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} \]Thus, the distance from \(P\) to \(R\) is \(\sqrt{20}\).

Step 2: Calculate the Distance from Q to R

Using the same formula, find the distance from point \(Q(-1, 3)\) to point \(R(-1, -1)\):\[ d_{QR} = \sqrt{((-1) - (-1))^2 + ((-1) - 3)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{0 + 16} = \sqrt{16} \]Thus, the distance from \(Q\) to \(R\) is \(\sqrt{16} = 4\).

Step 3: Compare the Distances

We have calculated the distances:- Distance from \(P\) to \(R\) is \(\sqrt{20}\).- Distance from \(Q\) to \(R\) is \(4\).Since \(4 < \sqrt{20}\), \(Q\) is closer to \(R\) than \(P\).

Key Concepts

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. It involves understanding and solving problems related to points, lines, angles, surfaces, and solids. In this exercise, we explore the geometry of the coordinate plane to determine which point is closer to another point. The distance formula is a vital tool in geometry, allowing us to calculate the exact distance between two points on the plane. This formula is derived from the Pythagorean theorem, which is a fundamental principle in geometric calculations. By comprehensively understanding geometry and its elements like the distance formula, you can solve a wide variety of spatial problems.

Coordinate Plane

A coordinate plane is a two-dimensional surface where each point is defined by a pair of numbers, typically known as coordinates. These coordinates are represented as

• x-coordinate: Indicates horizontal position
• y-coordinate: Indicates vertical position

It consists of two perpendicular lines called axes: the horizontal axis (x-axis) and the vertical axis (y-axis). The point where these axes intersect is known as the origin, denoted by \((0, 0)\).
The coordinate plane is the foundation for analyzing geometric shapes and solving problems involving distance, midpoint, and slope. It gives a visual representation of algebraic equations and inequalities. In this problem, the coordinate plane helps identify the relative positions of the points, allowing us to apply the distance formula effectively.

Points on a Plane

Points on a plane are specific locations that are defined using a pair of coordinates in the coordinate system. A point is represented as \((x, y)\) where \(x\) and \(y\) indicate its horizontal and vertical positions, respectively. In this problem, the points \(P(3, 1)\), \(Q(-1, 3)\), and \(R(-1, -1)\) are plotted on the coordinate plane. We compare distances to assess which point is closer to \(R\).
Understanding how points are positioned helps visualize and solve geometry problems more effectively. Using techniques like the distance formula reveals hidden relationships between points, as in determining proximity or alignment. This becomes especially important in applications such as navigation, graphics programming, and data visualization.