Question

what does the distance formula calculate? a. the slope between two points. b. the angle between two lines. c. the midpoint between two points. d. the distance between two points. triangle a has coordinates (-6,2), (-7,4), and (-3,4). what are the new coordinates of triangle b if the triangle has been rotated 90 degrees counterclockwise and then reflected over the x = 0 line? a. (-4,3), (-4,7), and (-2,6)

Explanation:

Step1: Recall distance - formula concept

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

Step1: Apply 90 - degree counter - clockwise rotation formula

The rule for a 90 - degree counter - clockwise rotation about the origin is $(x,y)\to(-y,x)$.
For point $(-6,2)$: $(-6,2)\to(-2,-6)$
For point $(-7,4)$: $(-7,4)\to(-4,-7)$
For point $(-3,4)$: $(-3,4)\to(-4,-3)$

Step2: Apply reflection over the line $x = 0$ (y - axis)

The rule for reflection over the y - axis is $(x,y)\to(-x,y)$.
For $(-2,-6)$: $(-2,-6)\to(2,-6)$
For $(-4,-7)$: $(-4,-7)\to(4,-7)$
For $(-4,-3)$: $(-4,-3)\to(4,-3)$
However, there seems to be an error in the provided options as the correct new coordinates after the given transformations are $(2,-6),(4,-7),(4,-3)$ and none of the provided options match. If we assume a possible mis - understanding in the problem setup or options, and re - check the work, based on the steps of transformation, we note that the steps are correct for the general transformation rules.

Answer:

d. The distance between two points.