Question

given a(8, 1) and b(2, -5), find the following:
a) the length of (overline{ab}) (exact answer)
b) the slop

Explanation:

Part (a): Length of \(\overline{AB}\)

Step1: Recall Distance Formula

The distance \(d\) 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 \(A(8, 1)\) and \(B(2, -5)\), \(x_1 = 8\), \(y_1 = 1\), \(x_2 = 2\), \(y_2 = -5\).

Step2: Substitute Values

Substitute into the formula: \(d = \sqrt{(2 - 8)^2 + (-5 - 1)^2}\)

Step3: Simplify Inside the Square Root

Calculate \((2 - 8)^2 = (-6)^2 = 36\) and \((-5 - 1)^2 = (-6)^2 = 36\). Then \(d = \sqrt{36 + 36}\)

Step4: Combine and Simplify

\(36 + 36 = 72\), so \(d = \sqrt{72}\). Simplify \(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\)

Step1: Recall Slope Formula

The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For \(A(8, 1)\) and \(B(2, -5)\), \(x_1 = 8\), \(y_1 = 1\), \(x_2 = 2\), \(y_2 = -5\).

Step2: Substitute Values

Substitute into the formula: \(m = \frac{-5 - 1}{2 - 8}\)

Step3: Simplify Numerator and Denominator

Numerator: \(-5 - 1 = -6\), Denominator: \(2 - 8 = -6\)

Step4: Divide

\(m = \frac{-6}{-6} = 1\)

Answer:

\(6\sqrt{2}\)

Part (b): Slope of \(\overline{AB}\)