A Cartesian coordinate system was used to identify locations on a circular track. As shown in Figure 1.4, the circular track contains the points A(-2,-4), B(-2, 3), C(5, 2). Find the total length of the tracks
Answer (1)
Assumptions: 1. The points form a reasonable arc of the circle: The points A(-2,-4), B(-2, 3), and C(5,2) are not collinear (they don't lie on a straight line). We assume they represent a significant portion of the circle's circumference.2. The circle is reasonably well-defined by these points: The points are not so widely spaced that multiple circles could pass through them. Solution: 1. Find the distances between the points: We use the distance formula: d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}- Distance AB: d_{AB} = \sqrt{(-2 - (-2))^2 + (3 - (-4))^2} = \sqrt{0^2 + 7^2} = 7- Distance BC: d_{BC} = \sqrt{(5 - (-2))^2 + (2 - 3)^2} = \sqrt{7^2 + (-1)^2} = \sqrt{50}- Distance AC: d_{AC} = \sqrt{(5 - (-2))^2 + (2 - (-4))^2} = \sqrt{7^2 + 6^2} = \sqrt{85}2. Estimate the radius: Since we only have three points, we can't precisely determine the circle's radius and center. However, we can make a reasonable estimate. The points suggest a circle with a radius roughly between 4 and 5 units. Let's assume a radius of 4.5 for our calculation.3. Calculate the circumference: The circumference of a circle is given by C = 2\pi r. Using our estimated radius:C = 2\pi (4.5) \approx 28.27 units. Therefore, an estimate of the total length of the track is approximately 28.27 units. This is an approximation because we had to estimate the radius based on limited information. A more precise solution would require the image to determine the circle's center and radius accurately.
