Question

for which of these triangles is the incenter also the circumcenter?
scalene triangle
right triangle
equilateral triangle
obtuse triangle

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

what is the point called where the perpendicular bisectors of a triangle intersect?
tangent
centroid
incenter
circumcenter

which circle is centered at the incenter of a triangle?
tangent circle
circumcircle
incircle
concentric circle

what is the term for a line segment that joins a vertex of a triangle to the midpoint of the opposite side?
angle bisector
tangent
midpoint
median

in which triangle do the centroid and circumcenter lie at the same location?
scalene triangle
right triangle
isosceles triangle
equilateral triangle

what is the relationship between the centroid and the vertices of a triangle?
it is equidistant from all vertices
it is two - thirds of the distance from each vertex to the midpoint of the opposite side

Explanation:

• For the first question: In an equilateral triangle, the in - center (center of the inscribed circle, intersection of angle bisectors) and the circum - center (center of the circumscribed circle, intersection of perpendicular bisectors) coincide.
• For the second question: In a right - triangle, the circum - center is located at the mid - point of the hypotenuse. This is because the hypotenuse is the diameter of the circumscribed circle.
• For the third question: The circum - center is defined as the point where the perpendicular bisectors of a triangle intersect.
• For the fourth question: The in - center is the center of the incircle (the largest circle that can be inscribed within the triangle).
• For the fifth question: A median of a triangle is a line segment that joins a vertex of a triangle to the mid - point of the opposite side.
• For the sixth question: In an equilateral triangle, the centroid (intersection of medians) and the circum - center (intersection of perpendicular bisectors) lie at the same location.
• For the seventh question: The centroid divides each median in a ratio of 2:1. It is two - thirds of the distance from each vertex to the mid - point of the opposite side.

Answer:

1. C. Equilateral triangle
2. B. It is located at the midpoint of the hypotenuse
3. D. Circumcenter
4. C. Incircle
5. D. Median
6. D. Equilateral triangle
7. B. It is two - thirds of the distance from each vertex to the midpoint of the opposite side