Problem 120 - Elementary and Intermediate Algebra

Question

Consider the line that passes through \(P(-2,3)\) and \(Q(4,-4)\). Find the distance between \(P\) and \(Q\).

Step-by-Step Solution

Answer

The distance between \(P\) and \(Q\) is \(\sqrt{85}\), approximately 9.22.

Step 1: Identify the coordinates of the points

The points given are \(P(-2, 3)\) and \(Q(4, -4)\). Each point has an \(x\)-coordinate and a \(y\)-coordinate.

Step 2: Recall the distance formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane is given by the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]where \((x_1, y_1) = (-2, 3)\) and \((x_2, y_2) = (4, -4)\).

Step 3: Substitute the coordinates into the formula

Substitute the coordinates of points \(P\) and \(Q\) into the distance formula:\[d = \sqrt{(4 - (-2))^2 + (-4 - 3)^2}\]Simplify inside the parentheses:\[d = \sqrt{(4 + 2)^2 + (-4 - 3)^2}\]

Step 4: Simplify the expression

Calculate inside the parentheses:\[d = \sqrt{6^2 + (-7)^2}\]Compute the squares:\[d = \sqrt{36 + 49}\]Add the results:\[d = \sqrt{85}\]

Step 5: Calculate the final result

\(\sqrt{85}\) is the distance between points \(P\) and \(Q\). This will be approximately 9.22, when calculated, but can be left as \(\sqrt{85}\) for an exact value.

Key Concepts

Coordinate Geometry

Coordinate Geometry is an essential aspect of mathematics that connects algebra and geometry, typically involving the use of coordinates to study geometrical shapes. It uses an ordered pair of numbers, usually in the form \((x, y)\), on a coordinate plane to represent points.These coordinates are crucial to locate points and can effortlessly describe lines, circles, and other geometrical shapes.
The importance of Coordinate Geometry lies in its ability to:

• Provide a systematic way to analyze geometric shapes and their properties.
• Allow solving problems by simulating conditions using equations.
• Enable transition from physical real-world scenarios to abstract mathematical models.

You encounter it in real life through maps, computer graphics, and even designing objects. So understanding Coordinate Geometry helps in visualizing how these mathematical concepts take shape in the world, making mathematics a powerful tool for solving practical problems.

Euclidean Distance

Euclidean Distance is a fundamental metric in mathematics used to determine the straight-line distance between two points in a plane. It is named after the ancient Greek mathematician Euclid, who first formulated many of the ideas that describe this concept. It is often visualized as the "as-the-crow-flies" distance that connects two points directly without any detours.
To calculate the Euclidean Distance between two points, use the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula captures the core idea that in a right triangle, the distance between two points is akin to the hypotenuse.
By finding the difference in the x and y coordinates (\(x_2 - x_1\) and \(y_2 - y_1\) respectively) and squaring these differences, one can form the two legs of a right triangle. Adding these together and taking the square root results in the straight-line or 'direct' distance between the two points.
Euclidean Distance is applicable in numerous fields such as physics, computer science (especially in algorithms and data clustering), and daily activities like navigating using GPS.

Coordinate Plane

The Coordinate Plane is a two-dimensional surface defined by a horizontal number line called the x-axis and a vertical number line called the y-axis. These axes intersect at a point called the origin, represented by the coordinates \((0, 0)\). This setup allows for a systematic way of describing a point in space using an ordered pair \((x, y)\), which represents how far the point lies from the origin.
The significance of the Coordinate Plane includes:

• Facilitating a visual representation of algebraic equations and geometrical figures.
• Enabling a clear visualization of operations like addition, subtraction, and transformations involving points.
• Helping to perform analysis and identify relationships between different geometric objects easily.

As you work with the Coordinate Plane, you gain insight into the spatial relationships and patterns, enhancing both problem-solving skills and spatial awareness. This system is prevalent in various scenarios, from simple graphing tasks in school mathematics to complex tasks such as engineering designs and game development grid systems.