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 mid - point of the medians c. the in - center is the mid - point 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:

Step1: 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 \(x^{2}+y^{2}=r^{2}\).

Step2: Solve problem 17

For a circle with center at the origin \((0,0)\) and radius \(r = 10\), substituting into the formula \(x^{2}+y^{2}=r^{2}\), we get \(x^{2}+y^{2}=100\). So the answer to 17 is d.

Step3: Recall in - center definition

The in - center of a triangle is the center of the inscribed circle (incircle). It is the point of intersection of the angle bisectors of the triangle. So the answer to 18 is a.

Step4: Solve problem 19

The equation of the circle is \((x - 10)^2+(y + 3)^2=1\). Comparing with the standard form \((x - h)^2+(y - k)^2=r^2\), we have \(r^{2}=1\), so \(r = 1\). The answer to 19 is d.

Answer:

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