Question

find the distance ( d(p_1,p_2) ) between the given points ( p_1 ) and ( p_2 ).
( p_1=(1,1) )
( p_2=(3,5) )
( d(p_1,p_2)=square )
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Identify the coordinates

Let \(P_1=(x_1,y_1)=(1,1)\) and \(P_2=(x_2,y_2)=(3,5)\)

Step2: Use the distance formula

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
Substitute the values: \(d=\sqrt{(3 - 1)^2+(5 - 1)^2}\)

Step3: Simplify the expression

First, calculate \((3 - 1)^2=2^2 = 4\) and \((5 - 1)^2=4^2 = 16\)
Then \(d=\sqrt{4 + 16}=\sqrt{20}\)
Simplify \(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\)

Answer:

\(2\sqrt{5}\)