Question

1. which of the following represents a circle with its center at the origin and a radius of 10? a. x² - y² = 100 b. (x + 10)² + (y + 10)² = 100 c. x² + y² = 10 d. x² + y² = 100 18. what is the significance of the in - center in constructing an incircle? a. the in - center is the center of the incircle b. the in - center is the midpoint of the medians c. the in - center is the midpoint of the sides d. the in - center is the center of the circumcircle 19. what is the radius of the circle with the equation (x - 10)² + (y + 3)² = 1? a. 10 b. 2 c. 0.5 d. 1

Explanation:

17. Recall circle - equation formula

The standard form of the equation of a circle with center \((h,k)\) and radius \(r\) is \((x - h)^2+(y - k)^2=r^2\). When the center is at the origin \((0,0)\), the equation is \(x^{2}+y^{2}=r^{2}\). Given \(r = 10\), then \(r^{2}=100\), so the equation is \(x^{2}+y^{2}=100\).

18. Understand in - center concept

The in - center of a triangle is the point of intersection of the angle bisectors of the triangle, and it is the center of the inscribed circle (incircle).

19. Identify radius from circle equation

For the circle equation \((x - 10)^2+(y + 3)^2=1\), comparing with the standard form \((x - h)^2+(y - k)^2=r^2\), we have \(r^{2}=1\), so \(r = 1\).

Answer:

1. d. \(x^{2}+y^{2}=100\)
2. a. The in - center is the center of the incircle
3. d. 1