Question

1. if a quadrilateral is inscribed in a circle, how do the opposite angles behave? a. they are complementary b. they are supplementary c. they are equal d. they are neither complementary nor supplementary 9. which set of points has an undefined slope? a. (6,5) and (7,5) b. (2,3) and (-2,4) c. (8,1) and (8,8) d. (4,5) and (5,8) 10. what type of triangle is used to derive the distance formula from the pythagorean theorem? a. equilateral triangle b. scalene triangle c. isosceles triangle d. right triangle 11. how do you represent the distance between points (x1,y1) and (x2,y2)? a. (x2 - x1)-(y2 - y1) b. $sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ c. (x1 + x2)+(y1 + y2) d. (x2 - x1)+(y2 - y1) 12. for which purpose is the distance formula particularly important in robotics? a. sensor tuning b. measuring between points c. power management d. signal processing 13. if two lines in a coordinate plane do not intersect, they are necessarily a. perpendicular b. parallel c. non - linear d. concurrent 14. what is the slope of a line perpendicular to another line with a slope of 2? a. -$\frac{1}{2}$ b. 2 c. - 2 d. $\frac{1}{2}$

Explanation:

1. For the question about opposite angles in a quadrilateral inscribed in a circle, the property is that opposite angles of a cyclic quadrilateral are supplementary.
2. The slope between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. If the denominator $x_2 - x_1=0$, the slope is undefined. For points with the same x - coordinate, the slope is undefined.
3. The distance formula is derived from the Pythagorean theorem using a right - triangle.
4. The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
5. In robotics, the distance formula is important for measuring between points.
6. In a coordinate plane, if two lines do not intersect, they are parallel.
7. The slope of a line perpendicular to a line with slope $m$ is $-\frac{1}{m}$. If $m = 2$, the slope of the perpendicular line is $-\frac{1}{2}$.

Answer:

1. b. They are supplementary
2. d. $(5,5)$ and $(5,8)$ (since they have the same x - coordinate, slope is undefined)
3. d. Right triangle
4. b. $\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$
5. b. Measuring between points
6. b. Parallel
7. a. $-\frac{1}{2}$