Question

1. the point where the perpendicular bisectors of a triangle intersect is called the: a. centroid b. circumcenter c. orthocenter d. incenter 11. what is required to uniquely determine the tangent lines to a circle from an external point? a. the circles diameter b. the distance to the nearest point on the circle c. just the circles radius d. the external point and the circles center 12. what is the significance of the incenter in constructing an incircle? a. the incenter is the midpoint of the sides b. the incenter is the midpoint of the medians c. the incenter is the center of the circumcircle d. the incenter is the center of the incircle

Explanation:

1. The circum - center of a triangle is defined as the point of intersection of the perpendicular bisectors of the sides of the triangle.
2. To uniquely determine the tangent lines to a circle from an external point, we need the external point and the center of the circle. Using the properties of tangents (tangent is perpendicular to the radius at the point of tangency), with the external point and the center, we can construct the tangents.
3. The in - center of a triangle is the center of the incircle (the largest circle that can be inscribed within the triangle). It is the point of intersection of the angle bisectors of the triangle.

Answer:

1. b. Circumcenter
2. d. The external point and the circle's center
3. d. The incenter is the center of the incircle