Question

what is the relationship between the centroid and the vertices of a triangle?
a. it is equidistant from all vertices
b. it is two - thirds of the distance from each vertex to the midpoint of the opposite side
c. it lies on the circumcircle
d. it forms right angles with the base of the triangle
what is the key property of the perpendicular bisector in a triangle?
a. they are always parallel to the sides of the triangle.
b. they are equidistant from all sides of the triangle.
c. they always intersect at the centroid.
d. they divide each side into equal halves.
which of the following points is always inside any triangle?
a. circumcenter
b. orthocenter
c. tangent
d. incenter
which point is equidistant from all vertices of a triangle?
a. incenter
b. circumcenter
c. centroid
c. tangent

Explanation:

• First question: The centroid of a triangle divides each median in a ratio of \(2:1\), meaning it is two - thirds of the distance from each vertex to the midpoint of the opposite side.
• Second question: The perpendicular bisector of a side of a triangle divides the side into two equal halves.
• Third question: The incenter is the center of the incircle of a triangle and is always inside the triangle. The circumcenter can be outside (for obtuse triangles), the orthocenter can be outside (for obtuse triangles), and a tangent is a line, not a point inside the triangle.
• Fourth question: The circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle and is equidistant from all the vertices of the triangle. The incenter is equidistant from the sides, the centroid has the \(2:1\) ratio property with medians.

Answer:

1. B. It is two - thirds of the distance from each vertex to the midpoint of the opposite side
2. D. They divide each side into equal halves
3. D. Incenter
4. B. Circumcenter