Question

homework 5.3

1. find the distance between the two points and simplify the radical.
2. (-1, 0), (-3, -4)

graph of a coordinate plane with a line segment between two points

Explanation:

Step1: Recall Distance Formula

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

Step2: Identify Coordinates

Let \((x_1, y_1)=(-1, 0)\) and \((x_2, y_2)=(-3, -4)\).

Step3: Substitute into Formula

Substitute the values into the formula:
\(d = \sqrt{(-3 - (-1))^2 + (-4 - 0)^2}\)
Simplify inside the square root:
\(-3 - (-1)= -3 + 1 = -2\), \(-4 - 0 = -4\)
So \(d = \sqrt{(-2)^2 + (-4)^2}\)

Step4: Simplify the Radicand

Calculate the squares: \((-2)^2 = 4\), \((-4)^2 = 16\)
Then \(d = \sqrt{4 + 16}=\sqrt{20}\)

Step5: Simplify the Radical

Simplify \(\sqrt{20}\) by factoring out the perfect square: \(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\)

Answer:

\(2\sqrt{5}\)