Question

1. which unit below is used to measure angles in circles? a. radians b. meters c. inches d. acres 21. the in - center of a triangle is found at the intersection of the: a. altitudes b. perpendicular bisectors c. medians d. angle bisectors 22. what is the length of the radius of a circle with the general form equation $x^{2}+y^{2}-4x + 6y+4 = 0$? a. 2 b. 6 c. 3 d. 4

Explanation:

Step1: Recall angle - measurement units

Radians are a unit for measuring angles in circles. Meters and inches are units of length, and acres are a unit of area. So, for the first question, the answer is radians.

Step2: Recall the definition of the in - center

The in - center of a triangle is the point of intersection of the angle bisectors of the triangle. Altitudes intersect at the ortho - center, perpendicular bisectors at the circum - center, and medians at the centroid.

Step3: Convert the circle equation to standard form

The general form of a circle equation is \(x^{2}+y^{2}-4x + 6y+4 = 0\). We complete the square for \(x\) and \(y\) terms.
For the \(x\) terms: \(x^{2}-4x=(x - 2)^{2}-4\).
For the \(y\) terms: \(y^{2}+6y=(y + 3)^{2}-9\).
The equation becomes \((x - 2)^{2}-4+(y + 3)^{2}-9 + 4=0\), which simplifies to \((x - 2)^{2}+(y + 3)^{2}=9\). The standard form of a circle equation is \((x - a)^{2}+(y - b)^{2}=r^{2}\), where \(r\) is the radius. Here \(r^{2}=9\), so \(r = 3\).

Answer:

1. A. Radians
2. D. Angle bisectors
3. C. 3