Problem 35 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

Point Q is closer to point R.

Step 1: Understand the Distance Formula

The distance formula helps find the distance between two points in a coordinate plane. The formula is: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). This formula will be used to calculate the distance from point \(R(-1,-1)\) to points \(P(3,1)\) and \(Q(-1,3)\).

Step 2: Calculate Distance from Point P to Point R

Use the distance formula with \(P(3,1)\) and \(R(-1,-1)\). Substitute the coordinates into the formula:\[d_{PR} = \sqrt{(3 - (-1))^2 + (1 - (-1))^2}\] This simplifies to: \[d_{PR} = \sqrt{(4)^2 + (2)^2} = \sqrt{16 + 4} = \sqrt{20}\] Calculated distance: \(\sqrt{20}\).

Step 3: Calculate Distance from Point Q to Point R

Use the distance formula with \(Q(-1,3)\) and \(R(-1,-1)\). Substitute the coordinates into the formula:\[d_{QR} = \sqrt{((-1) - (-1))^2 + (3 - (-1))^2}\] This simplifies to: \[d_{QR} = \sqrt{0 + 4^2} = \sqrt{16}\] Calculated distance: \(4\).

Step 4: Compare the Distances

Compare the calculated distances: \(d_{PR} = \sqrt{20} \approx 4.47\) and \(d_{QR} = 4\). Since \(4 < 4.47\), point \(Q(-1,3)\) is closer to point \(R(-1,-1)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, often called analytic geometry, combines algebra and geometry to study points, lines, and shapes on a coordinate plane. The plane consists of two axes -- the x-axis (horizontal) and the y-axis (vertical).
By plotting points using pairs of coordinates, such as \(3,1\) or \(-1,3\), we can map positions and analyze geometrical shapes. These coordinates consist of an ordered pair \(x, y\) that determines a point's location.
This approach creates a way to apply numerical methods and equations to solve geometric problems. So instead of purely abstract reasoning, we rely on coordinates and formulas to find distances, angles, and intersections. This concept serves as the backbone for many geometric calculations.

Distance Calculation

One crucial application of coordinate geometry is calculating the distance between two points. This is done using the distance formula. The formula originates from the Pythagorean theorem. It allows us to compute the length of the line segment connecting two points \( (x_1, y_1) \) and \( (x_2, y_2) \).
The formula is simple: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]\ This equation calculates the square root of the sum of the squares of the differences in the points' coordinates.
Consider our example with point \( R(-1, -1) \). By applying the formula to point \( P(3, 1) \), we find the distance \(d_{PR} = \sqrt{20} \). Conversely, for point \( Q(-1, 3) \), we discover that \(d_{QR} = 4 \). Hence, knowing the formula and how to apply it can swiftly solve puzzles relating to distances in coordinate geometry.

Problem Solving Steps

Solving geometrical problems using coordinate geometry typically involves a structured approach. Follow these effective problem-solving steps:
- **Understand the problem statement**: Clearly define what you need to find, such as the distance or midpoint.
- **Identify the coordinates of the points involved**: Extract the \(x\) and \(y\) coordinates from the problem statement. This ensures you have accurate data for calculations.
- **Apply the appropriate formula**: Whether it be the distance, midpoint, or slope formulas, knowing which formula to use is essential.
- **Substitute the values and simplify**: Insert the coordinates into the formula and perform the necessary arithmetic operations. Simplifying makes the result clearer and easier to interpret.
- **Compare**: If solving multiple distances, like our points \(P(3,1)\) and \(Q(-1,3)\) to \(R(-1,-1)\), compare values to conclude which point is closer.
These structured steps turn complex problems into manageable tasks, enhancing comprehension and ensuring accuracy in geometric calculations.