Problem 2 - Precalculus Mathematics for Calculus

Question

The distance between the points 1a, b2 and 1c, d 2 is _________. So the distance between 11, 22 and 17, 102 is _______.

Step-by-Step Solution

Answer

10

Step 1: Identify the points

We are given two points: \((x_1, y_1)\) and \((x_2, y_2)\). According to the problem, these points are \((1, 2)\) and \((7, 10)\).

Step 2: Use the distance formula

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

Step 3: Substitute the coordinates into the formula

Substitute \(x_1 = 1\), \(y_1 = 2\), \(x_2 = 7\), and \(y_2 = 10\) into the formula: \[d = \sqrt{(7 - 1)^2 + (10 - 2)^2}\]

Step 4: Solve the expression inside the square root

Start by solving the expressions inside the square root: \((7 - 1)^2 = 6^2 = 36\) and \((10 - 2)^2 = 8^2 = 64\).

Step 5: Calculate the sum

Add the squared differences: \[36 + 64 = 100\]

Step 6: Find the square root

Take the square root of the sum to find the distance: \(\sqrt{100} = 10\).

Step 7: Finalize the answer

The distance between the points \((1, 2)\) and \((7, 10)\) is \(10\).

Key Concepts

Coordinate Geometry

This area of mathematics is all about understanding shapes and figures by using a coordinate system. The most common system is the Cartesian plane, where every point is described using a pair of coordinates

• The horizontal coordinate is known as the x-coordinate
• The vertical coordinate is known as the y-coordinate

A single point in this plane looks like \((x, y)\), with exact positions on the grid dictated by these coordinates. For example, in our exercise, the points \((1, 2)\) and \((7, 10)\) tell us exactly where these points are on the plane.
Knowing how to plot and understand these coordinates helps us in finding not only distances but angles, slopes of lines, and many other geometric properties. It is a crucial skill in mathematics that finds application in many real-world scenarios like navigation and computer graphics.

Distance Calculation

The distance between two points in a coordinate plane is an extension of the Pythagorean theorem, a fundamental principle in mathematics. To find this distance, we rely on the distance formula:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Here is what each part of the formula means:

• \((x_2 - x_1)\): measures the horizontal distance (difference in x-coordinates) between two points
• \((y_2 - y_1)\): measures the vertical distance (difference in y-coordinates) between two points
• By squaring these differences, we measure the total horizontal and vertical movements, ensuring no negative values

This provides the diagonal distance across both the horizontal and vertical axes, giving a direct 'as the crow flies' measurement. In our example, we calculated this to show the distance between the points as 10 units.

Square Root

The concept of a square root is fundamental for solving distance calculations, as it undoes the squaring of numbers. When applying the distance formula, the sum of the squares of differences is calculated first:
In our exercise: \[6^2 + 8^2 = 36 + 64 = 100\]
Finding the length of the diagonal involves determining the square root of this sum:

• The square root \( \sqrt{100} \) returns the original length of the right triangle's hypotenuse without the squared units
• It simplifies understanding by converting 'area-like' dimensions back to linear measurement

This final step gives us the essential distance across both axes, crucially turning the mathematical equation into practical, usable knowledge. Ensuring fluency with square roots not only helps in distance calculations, but also in various other mathematical computations, such as solving quadratic equations.