Question

1. find the exact distance between the points.

$(x_1, y_1) = (-9, 4)$ and $(x_2, y_2) = (-5, -6)$

Explanation:

Step1: Recall Distance Formula

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Here, \( x_1 = -9 \), \( y_1 = 4 \), \( x_2 = -5 \), \( y_2 = -6 \).

Step2: Substitute Values into Formula

Calculate \( x_2 - x_1 = -5 - (-9) = -5 + 9 = 4 \) and \( y_2 - y_1 = -6 - 4 = -10 \).

Step3: Square the Differences

Square the results: \( (4)^2 = 16 \) and \( (-10)^2 = 100 \).

Step4: Sum the Squares

Add the squared values: \( 16 + 100 = 116 \).

Step5: Take the Square Root

Take the square root of the sum: \( d = \sqrt{116} = \sqrt{4 \times 29} = 2\sqrt{29} \).

Answer:

The exact distance between the points \((-9, 4)\) and \((-5, -6)\) is \( 2\sqrt{29} \).