Question

calculate the distance between the points ( f = (-1, 1) ) and ( e = (2, -4) ) in the coordinate plane. give an exact answer (not a decimal approximation).

Explanation:

Step1: Recall the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1=-1,y_1 = 1,x_2=2,y_2=-4\).

Step2: Substitute the values into the formula

Substitute the values: \(d=\sqrt{(2-(-1))^2+(-4 - 1)^2}\).
First, calculate \((2-(-1))=(2 + 1)=3\) and \((-4-1)=-5\). Then the formula becomes \(d=\sqrt{3^2+(-5)^2}\).

Step3: Simplify the expression

Calculate \(3^2 = 9\) and \((-5)^2=25\). So \(d=\sqrt{9 + 25}=\sqrt{34}\).

Answer:

\(\sqrt{34}\)