Medians in Triangles - Interactive Activity

Each median of a triangle runs from a vertex to the midpoint of the opposite side, and the three medians meet at one point — the centroid. These activities let you reshape a triangle and watch the centroid split every median in the same 2 to 1 ratio, then use that ratio to solve for missing pieces.

Definitions and Main Ideas

A median of a triangle runs from a vertex to the midpoint of the opposite side. Every triangle has three medians, and they meet at the centroid — the center of gravity of the triangle. The centroid has a measuring property that drives most of this lesson: it sits two-thirds of the way along each median from the vertex, so the vertex-to-centroid piece is twice as long as the centroid-to-midpoint piece.

Vocabulary

Median of a Triangle: A segment that goes from a vertex of a triangle to the midpoint of the opposite side.
Midpoint: The point on a segment that divides it into two congruent segments.
Concurrent: Three or more lines or segments that intersect at a single point.
Centroid: The point where the three medians of a triangle meet; it is the center of gravity of the triangle and lies two-thirds of the distance from each vertex to the midpoint of the opposite side.

Main Ideas

A median connects a vertex to the midpoint of the opposite side, so it splits that side into two equal lengths.
A triangle has up to three medians, and they are concurrent at the centroid.
The centroid divides each median in a 2 to 1 ratio: the piece from the vertex to the centroid is twice the piece from the centroid to the midpoint, and the vertex piece is two-thirds of the whole median.
The midpoint of a side can be found from coordinates by averaging the endpoints, which locates the far end of a median on the coordinate plane.

Centroid Explorer

What to do: Drag any vertex to reshape the triangle. All three medians and the centroid move with it. Choose a median with the buttons to measure its two pieces: the piece from the vertex and the piece from the centroid to the midpoint. Watch how the two pieces compare with the whole median.

Drag an orange vertex handle to reshape the triangle. Every median, midpoint, and the centroid are recomputed from the vertices.

These buttons choose which median the readout measures.

The matching tick marks on each side show that the midpoint splits the side into two congruent (equal in length) pieces. Sides with different lengths carry different numbers of ticks.

Centroid Challenge

What to do: Each problem shows a triangle with one median and its centroid. Use the centroid ratio and the midpoint to find the missing piece. Choose an answer. A first miss gives a hint and another try.

The Next item button cycles through all eight problems in order.

Build the Median

What to do: Work each problem one step at a time. First choose the statement or equation that governs the problem, then substitute the given values, then solve and answer what was asked. The diagram marks each piece only after its step is earned.

The Next problem button cycles through all six problems in order.

Your Turn

What to do: Answer each question. A first miss gives a hint and another try; the full explanation appears once a question is answered or after two misses. Only first-try answers count toward the score.

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Medians in Triangles — Mr. J Geometry. All activities on this page run entirely in your browser.