Question

1. which of the following sets of slopes represents perpendicular lines? a. 3 and -1/3 b. 0 and ∞ c. 1 and 1 d. 2 and -2
2. which pair of points would create a line with a slope of 0? a. (3,4) and (3,7) b. (1,1) and (2,2) c. (2,3) and (5,3) d. (1,1) and (1,2)
3. which theorem is the basis for deriving the distance formula? a. the midpoint theorem b. the circle theorem c. the pythagorean theorem d. the triangle inequality theorem

Explanation:

Step1: Recall slope - perpendicularity rule

The product of the slopes of two perpendicular lines is - 1. Also, a horizontal line (slope = 0) and a vertical line (undefined slope, often represented as $\infty$) are perpendicular.
For option a, $3\times(-\frac{1}{3})=- 1$.
For option b, a line with slope 0 is horizontal and a line with an undefined slope (represented as $\infty$) is vertical, so they are perpendicular.
For option c, $1\times1 = 1
eq - 1$.
For option d, $1\times(-2)=-2
eq - 1$. So, options a and b represent perpendicular lines.

Step2: Recall slope formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
For option a of question 11, $x_1 = 3,y_1 = 4,x_2 = 3,y_2 = 7$, then $m=\frac{7 - 4}{3 - 3}=\frac{3}{0}$, undefined.
For option b, $x_1 = 1,y_1 = 1,x_2 = 2,y_2 = 2$, then $m=\frac{2 - 1}{2 - 1}=1$.
For option c, $x_1 = 2,y_1 = 3,x_2 = 5,y_2 = 3$, then $m=\frac{3 - 3}{5 - 2}=0$.
For option d, $x_1 = 1,y_1 = 1,x_2 = 1,y_2 = 2$, then $m=\frac{2 - 1}{1 - 1}=\frac{1}{0}$, undefined.

Step3: Recall distance - formula derivation

The distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ is derived from the Pythagorean theorem. If we consider a right - triangle with the difference in x - coordinates as one leg and the difference in y - coordinates as the other leg, and the distance between two points as the hypotenuse.

Answer:

1. A. 3 and $-\frac{1}{3}$, B. 0 and $\infty$
2. C. $(2,3)$ and $(5,3)$
3. C. The Pythagorean Theorem