Triangle Centers and Their Properties

The video tutorial covers the four main centers of a triangle: incenter, centroid, orthocenter, and circumcenter. It explains how to find each center using specific geometric properties: the incenter through angle bisectors, the centroid through medians, the orthocenter through altitudes, and the circumcenter through perpendicular bisectors. The tutorial also discusses the positions of these centers in different types of triangles, such as acute, right, and obtuse triangles.

What is the incenter of a triangle?

• The point where the medians intersect

• The point where the angle bisectors intersect

• The point where the altitudes intersect

• The point where the perpendicular bisectors intersect

Which of the following is true about the incenter?

• It is always outside the triangle

• It is the center of the circumscribed circle

• It is the center of the inscribed circle

• It is the midpoint of the hypotenuse

How is the centroid of a triangle determined?

• By the intersection of the medians

• By the intersection of the perpendicular bisectors

• By the intersection of the altitudes

• By the intersection of the angle bisectors

What is a unique property of the centroid?

• It is always located at the right angle

• It divides each median into two segments, one twice the length of the other

• It is the center of the circumscribed circle

• It lies outside the obtuse triangle

Where is the orthocenter located in an acute triangle?

• Outside the triangle

• On the hypotenuse

• At the midpoint of the longest side

• Inside the triangle

In a right triangle, where is the orthocenter found?

• Inside the triangle

• Outside the triangle

• At the right angle

• On the hypotenuse

How is the circumcenter of a triangle determined?

• By the intersection of the perpendicular bisectors

• By the intersection of the angle bisectors

• By the intersection of the altitudes

• By the intersection of the medians

Where is the circumcenter located in a right triangle?

• At the right angle

• Outside the triangle

• On the hypotenuse

• Inside the triangle

What is the relationship between the circumcenter and the circumscribed circle?

• The circumcenter is always at the right angle

• The circumcenter is always inside the triangle

• The circumcenter is the center of the circumscribed circle

• The circumcenter is the center of the inscribed circle

Which triangle center is found by the intersection of the altitudes?

• Incenter

• Centroid

• Orthocenter

• Circumcenter