Triangle Theorems and Properties

This lesson covers various geometric theorems related to triangles, including perpendicular and angle bisector theorems, circumcenter, incenter, centroid, orthocenter, mid-segment theorem, and triangle inequalities. It explains how to find equations for perpendicular bisectors and provides examples to illustrate the hinge theorem. The lesson emphasizes understanding the properties and relationships within triangles.

What does the perpendicular bisector theorem state about a point on the perpendicular bisector of a segment?

• It forms an acute angle with the segment.

• It lies on the segment itself.

• It is closer to one endpoint than the other.

• It is equidistant to the endpoints of the segment.

If a point is equidistant from the endpoints of a segment, what can be inferred about its position?

• It lies on the perpendicular bisector of the segment.

• It lies on the segment itself.

• It is closer to one endpoint than the other.

• It forms an obtuse angle with the segment.

What does the angle bisector theorem state about a point on the angle bisector?

• It forms an acute angle with the sides.

• It is closer to one side than the other.

• It is equidistant to the sides of the angle.

• It lies on the angle itself.

What is the circumcenter of a triangle?

• The point where the three medians intersect.

• The point where the three altitudes intersect.

• The point where the three angle bisectors intersect.

• The point where the three perpendicular bisectors intersect.

Where does the circumcenter lie in an acute triangle?

• Inside the triangle

• On the triangle

• Outside the triangle

• At the midpoint of the hypotenuse

What is the centroid of a triangle?

• The point where the three angle bisectors intersect.

• The point where the three medians intersect.

• The point where the three perpendicular bisectors intersect.

• The point where the three altitudes intersect.

What is the orthocenter of a triangle?

• The point where the three altitudes intersect.

• The point where the three medians intersect.

• The point where the three perpendicular bisectors intersect.

• The point where the three angle bisectors intersect.

What does the mid-segment theorem state about the mid-segment of a triangle?

• It bisects the third side and is twice its length.

• It is parallel to the third side and half its length.

• It forms an acute angle with the third side.

• It is perpendicular to the third side and equal in length.

According to the triangle inequality theorem, what must be true for any three sides of a triangle?

• The sum of any two sides must be greater than the third side.

• The sum of any two sides must be twice the third side.

• The sum of any two sides must be less than the third side.

• The sum of any two sides must be equal to the third side.

What does the hinge theorem state about two triangles with two pairs of congruent sides and different included angles?

• The triangle with the smaller included angle has the longer third side.

• Both triangles have equal third sides.

• The triangle with the larger included angle has the longer third side.

• The third sides are proportional to the included angles.