Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the distance between the points \(P_{1}=(4,-7)\) and \(P_{2}=(-1,5)\).

Short answer

The distance between the points is 13.

Step-by-step solution

Identify the coordinates

The coordinates of the points are given. Point 1 is \( P_{1} = (4, -7) \) and Point 2 is \( P_{2} = (-1, 5) \).

Recall the distance formula

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

Substitute the coordinates into the formula

Substitute \( x_1 = 4 \), \( y_1 = -7 \), \( x_2 = -1 \), and \( y_2 = 5 \) into the distance formula: \[ d = \sqrt{((-1) - 4)^2 + (5 - (-7))^2} \]

Simplify the expressions inside the square root

Calculate the differences and their squares: \[ (-1 - 4) = -5 \rightarrow (-5)^2 = 25 \] \[ (5 - (-7)) = 12 \rightarrow (12)^2 = 144 \]

Add the squared differences

Add the squared differences: \[ 25 + 144 = 169 \]

Take the square root of the sum

Find the square root of 169: \[ \sqrt{169} = 13 \]

Key concepts

coordinate geometry

Coordinate geometry, also known as analytic geometry, connects algebra and geometry. It uses a coordinate system to describe the location of points on a plane. The most common coordinate system is the Cartesian coordinate system. Here, every point is represented by an ordered pair of numbers, \(x\) and \(y\), known as coordinates. The \(x\) value shows the point's horizontal position, while the \(y\) value shows the vertical position.

This system helps us solve geometric problems algebraically. By using coordinates, we can calculate distances, slopes, and midpoints, and even define shapes with equations. For instance, in our exercise, we identified two points, \(P_1 = (4, -7) \) and \(P_2 = (-1, 5) \). These points' coordinates help us determine the straight-line distance between them, making abstract geometry problems more tangible and calculable.

distance between points

The distance between two points in a coordinate plane tells us how far apart they are. It's a length that can be measured using the distance formula derived from the Pythagorean Theorem. The distance formula is:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula might look complex, but it breaks down into simple steps:

• Find the difference between the \(x\) coordinates (\