Question

name
date
find the slope between each pair of points. then, using the pythagorean theorem, find the distance between each pair of points. you may use the graph to help.

1. $(-2, -3)(1, 1)$

grid image
slope:
distance:

1. $(-7, 5)(-2, -7)$

grid image
slope:
distance:

Explanation:

Problem 17: Points \((-2, -3)\) and \((1, 1)\)

Slope Calculation

Step1: Recall slope formula

The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Here, \(x_1 = -2\), \(y_1 = -3\), \(x_2 = 1\), \(y_2 = 1\).

Step2: Substitute values into slope formula

\(m = \frac{1 - (-3)}{1 - (-2)} = \frac{1 + 3}{1 + 2} = \frac{4}{3}\)

Step1: Find horizontal and vertical distances

Horizontal distance (run) \(= |x_2 - x_1| = |1 - (-2)| = 3\)
Vertical distance (rise) \(= |y_2 - y_1| = |1 - (-3)| = 4\)

Step2: Apply Pythagorean theorem

Let \(d\) be the distance. Then \(d^2 = (\text{run})^2 + (\text{rise})^2\)
\(d^2 = 3^2 + 4^2 = 9 + 16 = 25\)

Step3: Take square root

\(d = \sqrt{25} = 5\)

Step1: Recall slope formula

The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Here, \(x_1 = -7\), \(y_1 = 5\), \(x_2 = -2\), \(y_2 = -7\).

Step2: Substitute values into slope formula

\(m = \frac{-7 - 5}{-2 - (-7)} = \frac{-12}{-2 + 7} = \frac{-12}{5} = -\frac{12}{5}\)

Answer:

(Slope): \(\frac{4}{3}\)

Distance Calculation (Using Pythagorean Theorem)