Problem 32 - Precalculus with Limits

Question

In Exercises 27-38, find the distance between the points. \( (8, 5) \), \( (0, 20) \)

Step-by-Step Solution

Answer

The distance between the points (8, 5) and (0, 20) is 17.

Step 1: Identify coordinates

The given points are (8, 5) and (0,20). We can label them as follows: Point 1: x1=8, y1=5 , Point 2: x2=0, y2=20. Now we have our \(x_1\), \(y_1\), \(x_2\), and \(y_2\) values.

Step 2: Calculate the differences in x and y coordinates

We need to calculate the differences between the x and y coordinates. So, we compute \(x_2-x_1\) and \(y_2-y_1\). This turns out to be (0-8) = -8 for x and (20-5) = 15 for y.

Step 3: Calculate the square of the differences

In this step we will square the differences calculated in the last step: This gives us \((-8)^2 = 64\) and \(15^2 = 225\).

Step 4: Add the squares

We add the squares we found in our third step. So, \(64 + 225 = 289\).

Step 5: Take the square root

The last step in the calculation is to take the square root of the sum we found in the fourth step. The square root of 289 is 17.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a system that uses numerical values to represent geometric figures and analyze their properties.

In this field, the position of points on the plane is described using an ordered pair of numbers, commonly referred to as coordinates. The Cartesian coordinate system, devised by René Descartes, is the most widely used and consists of two lines intersecting at a right angle forming axes: the horizontal axis (x-axis) and the vertical axis (y-axis).

For example, in the exercise given, the coordinates (8, 5) and (0, 20) represent points in this system. The first number of the pair corresponds to the position on the x-axis, while the second number corresponds to the position on the y-axis. By plotting these points, we can visualize their location in a two-dimensional space and further use concepts such as the distance formula to determine how far apart they are.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. According to this theorem, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is often expressed as a^2 + b^2 = c^2 where c is the hypotenuse and a and b are the other two sides.

When finding the distance between two points on the coordinate plane, we can imagine a right triangle formed by the horizontal and vertical distances between the points, with the distance between the points serving as the hypotenuse. This is why the Pythagorean theorem is so crucial when working with coordinate geometry. It is the basis for the distance formula that allows us to calculate this hypotenuse, thereby finding the distance between two points.

Distance Formula

The distance formula is directly derived from the Pythagorean theorem and is used to determine the distance between two points in the coordinate plane. It is expressed as D = sqrt((x2 - x1)^2 + (y2 - y1)^2), where D is the distance between the points, (x1, y1) and (x2, y2) are the coordinates of the two points.

This formula basically tells us that to find the distance, we need to find the horizontal ( x ) and vertical ( y ) distances between the points, square them, add them together, and then take the square root of the sum.

Applying the Distance Formula

For the points given in our exercise, (8, 5) and (0, 20), following the formula we calculate the difference between the x-coordinates (0 - 8), and square it to get 64. Similarly, we take the difference between the y-coordinates (20 - 5), square it to get 225, and then add the results to get 289. Lastly, we find the square root of 289, which is 17, yielding the distance between the two points.