Problem 12 - Calculus with Concepts in Calculus

Question

Determine the distance between the given points. \((a+e, b+e)\) and \((c+e, d+e)\)

Step-by-Step Solution

Answer

Distance is \( \sqrt{(c-a)^2 + (d-b)^2} \).

Step 1: Understanding the Request

We need to find the distance between two points given in a coordinate plane. The points are \((a+e, b+e)\) and \((c+e, d+e)\). This will involve using the distance formula.

Step 2: Review the Distance Formula

The distance \(D\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Step 3: Substitute the Given Points into the Formula

In the formula, substitute \((x_1, y_1) = (a+e, b+e)\) and \((x_2, y_2) = (c+e, d+e)\). The expression becomes: \[D = \sqrt{((c+e) - (a+e))^2 + ((d+e) - (b+e))^2}\]

Step 4: Simplify the Problem

Subtract \((a+e)\) from \((c+e)\) to simplify: \(c+e-a-e = c-a\)Likewise, subtract \((b+e)\) from \((d+e)\) to get:\(d+e-b-e = d-b\)Thus, the expression simplifies to: \[D = \sqrt{(c-a)^2 + (d-b)^2}\]

Step 5: Final Calculation

The expression simplifies to the typical distance formula without the extra \(e\) factor.The distance \(D\) is therefore: \[D = \sqrt{(c-a)^2 + (d-b)^2}\]

Key Concepts

Coordinate Geometry Basics

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe and analyze geometric figures. This approach connects geometric shapes to coordinate points in the plane. By translating shapes into numerical form, coordinate geometry equips us with the tools to apply algebraic methods to geometric problems. It allows us to calculate distances, areas, and angles precisely using formulas. One of the applications of coordinate geometry is to find the distance between points, which is fundamental in various fields such as navigation, physics, and engineering. Understanding this branch helps students solve problems visually and algebraically, making it a valuable aspect of mathematics learning.

Understanding Distance Calculation

Distance calculation in coordinate geometry involves finding the straight-line distance between two points on a plane. It uses the distance formula, which is derived from the Pythagorean theorem. Here's a refresher: the formula for the distance \(D\) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].

• The expression inside the square root represents the squared differences in the x-coordinates and y-coordinates of the points.
• The process begins with subtracting the x and y values of one point from those of the other point.
• Each of these resulting differences is then squared, added together, and finally, the square root of this sum is taken to find the distance.

This formula is a powerful tool for determining how far apart two points lie on a coordinate plane.

Exploring the Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numbers, known as coordinates. These numbers specifically indicate the point's location relative to the vertical y-axis and the horizontal x-axis. The plane comprises four quadrants, which help in identifying the sign and position of coordinates. For instance, the top-right quadrant contains points with positive x and y values.

• Any point on this plane is described using the format \( (x, y) \), where \(x\) is the horizontal position and \(y\) is the vertical position.
• Understanding how to navigate and interpret this plane is crucial for solving problems involving positions and distances.
• Knowing how changes in coordinates affect position helps in visualizing geometric transformations and in solving algebraic expressions tied to geometry.

Mastery of the coordinate plane is key to excelling in coordinate geometry and related problems.

Mathematics Problem Solving Strategies

Mathematics problem solving involves utilizing a combination of strategies and formulas to find solutions. When tackling problems in coordinate geometry, like finding the distance between points, it's important to follow a structured approach.

• Begin by identifying what is given and what needs to be found. In our exercise, the given are two points in coordinate form.
• Clearly write down the formula needed, in this case, the distance formula.
• Substitute the given values into the formula, carefully tracking each mathematical operation.
• Simplify and solve step-by-step to avoid mistakes and reach the correct solution.

This organized approach not only leads to the correct answer but also strengthens your mathematical reasoning and problem-solving skills. Practice in this area sharpens your ability to tackle diverse problems efficiently.