Centers of Triangles
The gravitational center of a triangle.
What is a centroid?
The point that is the intersection of all of the angle bisectors of a triangle is equidistant from the three sides.
What is the Incenter Theorem?
Two sides of a triangle are 6 and 20. What is the range of the third side?
14<x<26
Reason for step 2
What is the definition of a centroid?
In figure I, angle 5 must be greater than which angles?
Angles 1 and 2
The intersection of all three perpendicular bisectors of a triangle.
What is the circumcenter?
In triangle ABC, AB= 7, BC = 10, and AC = 8. Which is the largest angle?
Angle A.
Two sides of a triangle are 35 and 19. What is the range of the possible values of the third side?
What is the justification for step 3?
What is the definition of a median?
In Figure I angle 5 must be less than which angles?
Angles 8, 9, and 10
A line that goes from a vertex to the opposite side and is perpendicular to the opposite side.
What is the Hinge Theorem?
The three sides of a triangle are 5x, 7x, and 2x + 9. What is the range of values that x can equal?
x>9/10
What is the justification of step 6?
What is substitution?
In Figure 1, name all of the angles that angle 10 must be greater than.
Angles 9, 8, 7, 4, 5, 1, 2
The intersection of all of the angle bisectors of a triangle.
What is the incenter?
In triangle ABC AB = 5, and BC = 7. This theorem tells us that 2<AC<12.
What is the Triangle Inequality Theorem?
Three sides of a triangle are 19, 4x + 1, and 10x -5. What is the possible range of values for x?
23/4 < x < 25/6
What is the justification for step 4?
What is the definition of a midpoint?
In Figure II, if AC = EF, BC = ED, and AB > FD... what can you conclude?
angle C > angle E
In triangle ABC, the line connecting point A with the midpoint of segment BC intersects with the line connecting point B with the midpoint of segment AC at point D. If point D is 8 feet from point B, how far is point D from AC?
4 feet
In triangle ABC AB = 2, BC = 3, and AC = 2.6. In triangle EFG, EF = 2, FG = 3, and EG = 2.7. This theorem tells us that the measure of angle B is smaller than the measure of angle F.
What is the converse to the Hinge Theorem?
What is the justification for step 5?
What is the Triangle Inequality Theorem?
In Figure II, if J is a centroid and JC = 18, what is GC?
27