Question

question 2 of 5 select the correct answer from each drop - down menu. using the line segment shown, derive the distance formula.

Explanation:

Step1: Identify two points on the line

From the graph, we can identify two endpoints of the line segment. Let's assume the left - hand endpoint is \((x_1,y_1)=(-3, - 2)\) and the right - hand endpoint is \((x_2,y_2)=(4,4)\) (we can also use other pairs of points, but we will use these for demonstration).

Step2: Recall the Pythagorean theorem for distance

To find the distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\), we can form a right - triangle where the horizontal leg has length \(|x_2 - x_1|\) and the vertical leg has length \(|y_2 - y_1|\). By the Pythagorean theorem, \(d^{2}=(x_2 - x_1)^{2}+(y_2 - y_1)^{2}\), and then \(d = \sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\).

Let's calculate the horizontal difference: \(x_2 - x_1=4-(-3)=4 + 3=7\)

Let's calculate the vertical difference: \(y_2 - y_1=4-(-2)=4 + 2 = 6\)

Then, using the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\), we substitute \(x_2 - x_1 = 7\) and \(y_2 - y_1=6\) into the formula:

\(d=\sqrt{7^{2}+6^{2}}=\sqrt{49 + 36}=\sqrt{85}\)

(If we use the origin - related points, for example, if we consider the line passing through the origin \((0,0)\) and another point \((a,b)\), the distance from \((0,0)\) to \((a,b)\) is \(\sqrt{(a - 0)^{2}+(b - 0)^{2}}=\sqrt{a^{2}+b^{2}}\), which is a special case of the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\) when \((x_1,y_1)=(0,0)\))

Answer:

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