Circumcenter, Incenter, centroid, orthocenter

What is the relationship between the circumcenter and the circumcircle?

• The circumcenter is the center of the circumcircle, which is the circle that passes through all three vertices of the triangle.

• The circumcenter is the point where the altitudes of the triangle intersect.

• The circumcenter is the midpoint of the longest side of the triangle.

• The circumcenter is the point where the medians of the triangle intersect.

What is the centroid of a triangle?

• The point where the three medians of a triangle intersect.

• The point where the altitudes of a triangle intersect.

• The point that is equidistant from all three vertices of a triangle.

• The point that divides the triangle into two equal areas.

What is the significance of the centroid in a triangle?

• The centroid is the point where all three altitudes intersect.

• The centroid is the balance point of the triangle and divides each median into a 2:1 ratio.

• The centroid is the point where the three angle bisectors meet.

• The centroid is the midpoint of the longest side of the triangle.

What is the distance from the centroid to a vertex in relation to the median?

• 1/2 the length of the median

• 2/3 the length of the median

• 3/4 the length of the median

• Equal to the length of the median

If P is the incenter of triangle JKL, what does OP represent?

• The distance from the incenter to one of the sides of the triangle, which is the radius of the incircle.

• The length of the triangle's longest side.

• The area of triangle JKL.

• The perimeter of triangle JKL.

Define the circumcenter of a triangle.

• The circumcenter is the point where the three perpendicular bisectors of a triangle intersect. It is equidistant from all three vertices.

• The circumcenter is the point where the three altitudes of a triangle meet.

• The circumcenter is the center of the circle that can be inscribed within the triangle.

• The circumcenter is the midpoint of the longest side of the triangle.

What is the formula for finding the length of a median in a triangle?

• m = 1/2 * sqrt(2a^2 + 2b^2 - c^2)

• m = (a + b + c) / 3

• m = sqrt(a^2 + b^2)

• m = 1/2 * (a + b)

How can you determine if a triangle is acute, right, or obtuse based on its orthocenter?

• If the orthocenter is inside the triangle, it is acute.

• If the orthocenter is on the triangle, it is obtuse.

• If the orthocenter is outside the triangle, it is right.

• If the orthocenter is on the triangle, it is right.

What type of point of concurrency is the circumcenter?

• The point of concurrency of the medians of a triangle.

• The point of concurrency of the altitudes of a triangle.

• The point of concurrency of the perpendicular bisectors of a triangle.

• The point of concurrency of the angle bisectors of a triangle.

If G is the centroid of triangle ABC and AG = 16, what is the length of GD?

• 6

• 8

• 10

• 12