Question

20. circles o and p are tangent to the x-axis at (4, 0) and (13, 0), respectively. point a(4, 8) is on circle o and point b(13, 10) is on circle p. line segment op connects the center of circle o to the center of circle p. what is the slope of segment op? express your answer as a common fraction.

Answer 1

The slope of segment OP, connecting the centers of circles O and P tangent to the x-axis, is 0 as it is a horizontal line segment between points (4, 0) and (13, 0).

Step-by-step explanation:

The student is given two points that are centers of circles O and P, which are tangent to the x-axis and is asked to find the slope of segment OP. To find the slope of a line segment, we use the formula (y2 - y1) / (x2 - x1). Here, the coordinates of the centers can be derived from the tangent points and the points given on the respective circles. For circle O with tangent point (4, 0) and point A(4, 8), the center is at (4, 0), since the radius is perpendicular to the tangent at the point of tangency, which also implies that the radius is parallel to the y-axis. For circle P with tangent points (13, 0) and point B(13, 10), the center is similarly at (13, 0).

With centers O(4, 0) and P(13, 0), we have a horizontal line segment, OP, which means the rise is 0 and the run is 13 - 4. Hence, the slope of OP is 0 divided by 9, which is simply 0.