Distinguish medians, bisectors and altitudes, and recognize degenerate side data.
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Try
A triangle can have several useful centers. In the right triangle with vertices (0,0),(6,0),(0,6), the centroid is (2,2) while the circumcenter is (3,3). The diagram makes their separation visible; neither is simply “the center” without a definition.
Predict
Predict which construction answers each question. Medians meet at the centroid, which divides each median 2:1 from its vertex. Perpendicular bisectors meet at the circumcenter, equidistant from vertices. Internal angle bisectors meet at the incenter, equidistant from sides. Altitudes meet at the orthocenter. In an obtuse triangle, some centers lie outside it.
Experiment
Draw the midpoint of the hypotenuse in the diagram and join it to the opposite vertex. Locate the point two thirds along that median. Measure its coordinates using the stated scale, then verify them arithmetically. Separately test side triples 4,7,10 and 4,7,11 with strips or a sketch: equality at the largest side makes a straight configuration, not a triangle.
Right triangle with vertices (0,0), (6,0), (0,6), plotted with equal coordinate scales. G=(2,2) is the centroid; O=(3,3) is the circumcenter. They are different points.
Understand
Worked argument: for a nondegenerate triangle with two sides 6 and 10 and third side x, the strict triangle inequalities give x+6>10 and 6+10>x. Hence 4<x<16; the remaining inequality is then satisfied. Endpoints are excluded because they collapse the triangle into a line.
The circumcenter’s defining fact follows from a locus theorem: a point is equidistant from A,B exactly when it lies on their perpendicular bisector. Thus a point on the perpendicular bisectors of AB and AC has equal distances to all three vertices. Invalid proof: “All three bisectors look concurrent.” Equality of distances, not appearance, supplies the reasoning.
Apply
Name the center and the defining lines in each response. A median need not be perpendicular to its side, and an angle bisector need not bisect that side. When testing possible lengths, use strict inequalities and distinguish integer candidates from the full real interval.
Continue
Depth: derive the centroid coordinates by showing that the point averaging the three vertices divides each median in ratio 2:1. For circumcenters, use perpendicular-bisector equations; for a right triangle, the hypotenuse midpoint is equidistant from all vertices. The equalities coincide in an equilateral triangle, not in every triangle.
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Practice and reasoning
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A median has length 18 cm. How far is the centroid from its vertex, in cm?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Two sides are 5 and 9. How many integer lengths can the third side have in a nondegenerate triangle?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which lines meet at the circumcenter?
Which triple cannot be the sides of a nondegenerate triangle?
If O lies on the perpendicular bisectors of AB and AC, prove OA=OB=OC and explain why this locates a circle through A,B,C.
Structure your argument and preserve revisions
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Explain why a triangle's centroid and circumcenter need not coincide. Give an explicit right-triangle example.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
