Question 1: Explain why the equation of the line y=5 has a slope that is undefined. Explain in words or mathematically.
Question 2: Given any circle, the coordinates of one endpoint of a diameter is (3,-2). If the coordinates of the center are (2,-4), find the coordinates for the other endpoint.
Question 3: Explain what the difference is between a slope that is undefined and a slope of 0. Use mathematical reasoning to explain your answer.
THANKS!
I will be happy to critique your thinking.
Q1: Find the length and midpoint of segment PQ given P(3,2) and Q(-1,4)
Q2: The midpoint of AB is M(1/2 , 6). Given A(3,7), find the coordinates of B.
Question 1: The equation of the line y=5 represents a horizontal line that is parallel to the x-axis. The slope of a line is defined as the change in y-coordinates divided by the change in x-coordinates between any two points on the line. However, for a horizontal line like y=5, there is no change in y-coordinates because all the points on the line have the same y-value (which is 5 in this case). Therefore, the change in y-coordinates is always 0, and dividing by 0 is undefined in mathematics. So, the slope of the line y=5 is undefined.
Question 2: To find the coordinates of the other endpoint of the diameter, we need to use the fact that the center of the circle lies in the middle of the diameter. Since the center has coordinates (2,-4) and one endpoint is (3,-2), we can use the midpoint formula to find the coordinates of the other endpoint.
The midpoint formula states that the x-coordinate of the midpoint is the average of the x-coordinates of the two endpoints, and the same applies for the y-coordinate. Therefore, the x-coordinate of the other endpoint is the average of 3 and 2, which is (3 + 2) / 2 = 5 / 2 = 2.5.
Similarly, the y-coordinate of the other endpoint is the average of -2 and -4, which is (-2 + -4) / 2 = -6 / 2 = -3.
So, the coordinates of the other endpoint of the diameter are (2.5, -3).
Question 3: A slope of 0 means that the line is horizontal and parallel to the x-axis. It indicates that there is no change in the y-coordinates between any two points on the line. Mathematically, the change in y-coordinates divided by the change in x-coordinates results in 0, which is a defined value.
On the other hand, a slope that is undefined occurs when the line is vertical and parallel to the y-axis. In this case, there is no change in the x-coordinates between any two points on the line. Mathematically, the change in x-coordinates divided by the change in y-coordinates results in division by 0, which is undefined.
In summary, a slope of 0 indicates a horizontal line with no change in y-coordinates, while an undefined slope indicates a vertical line with no change in x-coordinates.