Question

in which triangle do the centroid and circumcenter lie at the same location?
a. scalene triangle
b. isosceles triangle
c. equilateral triangle
d. right triangle

which of the following describes the circumcenter in a right triangle?
a. it is located at the midpoint of the hypotenuse
b. it is the lowest point of the triangle
c. it coincides with the centroid
d. it divides the triangle into three smaller triangles of equal area

which of the following is true about the centroid?
a. it is equidistant from all vertices of the triangle.
b. it divides each median into a ratio of 2:1.
c. it is the center of the incircle.
d. it is equidistant from all sides of the triangle.

Explanation:

1. The centroid is the point of intersection of the medians of a triangle and the circum - center is the point of intersection of the perpendicular bisectors of the sides of a triangle. In an equilateral triangle, due to its symmetry, the centroid and circum - center coincide.
2. For a right - triangle, the circum - center is located at the mid - point of the hypotenuse because the hypotenuse is the diameter of the circum - circle.
3. The centroid of a triangle divides each median in the ratio 2:1 from the vertex.

Answer:

1. c. Equilateral triangle
2. a. It is located at the midpoint of the hypotenuse
3. b. It divides each median into a ratio of 2:1