Find the exact values for the lengths of the labeled segments a, b and p drawn in green, red, and blue, respectively. Note that r=3 is the radius of the circle, and s=2 is the arc length from the point (3,0) around the circle to the indicated point.
(So imagine a triangle drawn on the coordinate system in a circle. A and B are legs. P connects to the arc.)
No drawing. Cannot copy and paste here.
To find the exact values for the lengths of the labeled segments a, b, and p, you can use the geometry of the circle. Let's break down the steps to find each of these lengths:
1. Segment a:
- First, observe that segment a is a vertical line that connects point A to the x-axis.
- The x-coordinate of point A is equal to the radius of the circle, which is given as r = 3.
- Therefore, the length of segment a is equal to the x-coordinate of point A, which is a = 3.
2. Segment b:
- Segment b is a horizontal line that connects point B to the y-axis.
- To find the y-coordinate of point B, we need to consider the remaining distance after traversing the arc of length s = 2.
- The circumference of a circle is calculated as 2πr, where r is the radius. In this case, the radius is r = 3.
- The total circumference of the circle is 2π(3) = 6π.
- Since the arc length s = 2 is given, the remaining arc length after moving from point (3,0) around the circle will be 6π - 2.
- Since point B lies on the x-axis, the y-coordinate of point B will be 0.
- Therefore, the length of segment b is equal to the y-coordinate of point B, which is b = 0.
3. Segment p:
- Segment p is the line connecting point P to point B, extending from the arc.
- Since point P lies on the arc of the circle, we need to determine its coordinates.
- The arc length s = 2 is given, and it corresponds to an angle of (s/radians-of-a-full-circle) = (2/6π) of a full circle.
- Using this angle, we can calculate the x-coordinate of point P as r * cos(angle) = 3 * cos(2/6π).
- Similarly, we can calculate the y-coordinate of point P as r * sin(angle) = 3 * sin(2/6π).
- Therefore, the coordinates of point P are (3 * cos(2/6π), 3 * sin(2/6π)).
- To find the length of segment p, we can use the distance formula between points P and B, given their coordinates.
- The length of segment p is given by √((x2 - x1)² + (y2 - y1)²), where (x1, y1) = (3 * cos(2/6π), 3 * sin(2/6π)) and (x2, y2) = (0, 0).
- Plugging these values into the distance formula, we get p = √((0 - 3 * cos(2/6π))² + (0 - 3 * sin(2/6π))²).
Therefore, the exact values for the lengths of the labeled segments are:
Segment a: a = 3
Segment b: b = 0
Segment p: p = √((0 - 3 * cos(2/6π))² + (0 - 3 * sin(2/6π))²)