Question

1. find the slope between the points (5, 12) and (7, 16). use the slope formula.
2. given the following solve for x

2x + 8
6x - 20

1. describe the translation in word format for point p(3,-2). (x,y)→(x - 8,y + 4)
2. place the correct definition with each word.

angle____
parallel____
perpendicular____
circle____
segment____
a. two coplanar lines that are always equidistant apart and intersect.
b. two intersecting lines that form one 90 degree angle
c. two lines, segments or rays that share a common endpoint
d. a piece of a line between two endpoints
e. a set of points equidistant from a given point

Explanation:

Step1: Recall slope - formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $(x_1,y_1)=(5,12)$ and $(x_2,y_2)=(7,16)$.
$m=\frac{16 - 12}{7 - 5}$

Step2: Calculate the slope

$m=\frac{4}{2}=2$

Step3: Solve for x in parallel - lines problem

If the two lines are parallel, then the corresponding angles are equal, so $2x + 8=6x-20$.
First, move the x - terms to one side: $20 + 8=6x-2x$.
$28 = 4x$.
Then, solve for x: $x=\frac{28}{4}=7$

Step4: Describe the translation

For the point $P(3,-2)$ and the transformation $(x,y)\to(x - 8,y + 4)$.
The x - coordinate of the point changes from 3 to $3-8=-5$ (moves 8 units to the left), and the y - coordinate changes from - 2 to $-2 + 4=2$ (moves 4 units up). So the translation is 8 units to the left and 4 units up.

Step5: Match the definitions

• Angle: C. Two Lines, Segments or Rays that share a common Endpoint.
• Parallel: A. Two coplanar lines that are always equidistant apart and do not intersect.
• Perpendicular: B. Two intersecting lines that form one 90 - degree angle.
• Circle: E. A set of points equidistant from a given point.
• Segment: D. A piece of a line between two endpoints.

Answer:

1. The slope is 2.
2. $x = 7$
3. 8 units to the left and 4 units up.
4. Angle: C, Parallel: A, Perpendicular: B, Circle: E, Segment: D