Question

question 11: if y(1, 3) and a(5, 7), calculate ya in simplest radical form. show your algebraic thinking to earn full credit.

Explanation:

Step1: Recall distance formula

The distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For points \(Y(1,3)\) and \(A(5,7)\), \(x_1 = 1\), \(y_1 = 3\), \(x_2 = 5\), \(y_2 = 7\).

Step2: Substitute values into formula

Substitute the values into the distance formula: \(YA=\sqrt{(5 - 1)^2+(7 - 3)^2}\).

Step3: Simplify the expression

First, calculate the differences: \(5 - 1 = 4\) and \(7 - 3 = 4\). Then square them: \(4^2=16\) and \(4^2 = 16\). Add the results: \(16+16 = 32\). So \(YA=\sqrt{32}\).

Step4: Simplify the radical

Simplify \(\sqrt{32}\) by factoring 32 as \(16\times2\). Then \(\sqrt{32}=\sqrt{16\times2}=\sqrt{16}\times\sqrt{2}=4\sqrt{2}\).

Answer:

\(4\sqrt{2}\)