Question

find the distance between the points (7, 10) and (5, 6). write your answer as a whole number or a fully simplified radical expression. do not round.

units

Explanation:

Step1: Recall the distance formula

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \).
Here, \( x_1 = 7,y_1 = 10,x_2 = 5,y_2 = 6 \).

Step2: Substitute the values into the formula

Substitute the values into the formula:
\( x_2 - x_1=5 - 7=- 2 \), \( y_2 - y_1=6 - 10=-4 \)
Then \( d=\sqrt{(-2)^2+(-4)^2} \)

Step3: Simplify the expression inside the square root

First, calculate the squares: \((-2)^2 = 4\) and \((-4)^2=16\)
Then add them: \(4 + 16=20\)
So \( d=\sqrt{20} \)

Step4: Simplify the radical

Simplify \(\sqrt{20}\): we know that \(20 = 4\times5\), so \(\sqrt{20}=\sqrt{4\times5}=\sqrt{4}\times\sqrt{5}=2\sqrt{5}\)

Answer:

\(2\sqrt{5}\)