Centroid Theorems & Median Proofs

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Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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| Attempts: 21 | Questions: 20 | Updated: Jan 16, 2026
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1) If all three medians are drawn in a triangle, how many smaller triangles of equal area are formed?

Explanation

The three medians intersect at the centroid and divide the triangle into six smaller triangles of equal area.

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About This Quiz
Centroid Theorems & Median Proofs - Quiz

Want to see how the centroid can prove cool properties of triangles? In this quiz, you’ll apply what you know about medians to solve problems and explore proofs. You’ll see how the centroid divides each median into a 2:1 ratio, creates equal-area regions, and always stays inside the triangle. With... see moreeach question, you’ll grow more confident in using medians and centroids to explain why triangles work the way they do!
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2) The centroid divides a median so that the part closer to the vertex is:

Explanation

In every triangle, the vertex-to-centroid segment is twice as long as the centroid-to-midpoint segment.

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3) In △PQR, the medians divide the triangle into six regions of equal area. This property is a direct result of:

Explanation

Because the centroid divides each median in a 2:1 ratio, all six smaller triangles created by the medians have equal area.

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4) A median of a triangle is drawn from a vertex to ____.

Explanation

The centroid always divides each median in a 2:1 ratio, with the longer segment toward the vertex and the shorter toward the midpoint.

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5) In △ABC, medians AD, BE, and CF intersect at centroid G. Which of the following is always true?

Explanation

In a triangle, the centroid is located at the point where the three medians intersect, and it divides each median into a ratio of 2:1, with the longer segment being closer to the vertex.

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6) Which point of concurrency is the intersection of the medians?

Explanation

The centroid is the point where all three medians meet — the triangle’s center of balance.

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7) The centroid divides each median in the ratio ____.

Explanation

Every median is divided by the centroid so that the section from the vertex to centroid is twice as long as from the centroid to midpoint.

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8) If a median is 15 units long, the distance from the centroid to the midpoint of a side is ____.

Explanation

The centroid divides the median in a 2:1 ratio, so the shorter part is one-third of the total:

15 ÷ 3 = 5 units.

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9) The centroid is also called the ____.

Explanation

The centroid is often called the center of gravity or balancing point, because it’s where a triangle can balance perfectly on a pencil tip.

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10) In △XYZ, medians XP, YQ, and ZR intersect at centroid G. If XP=12, what are the lengths of XG and GP?

Explanation

Since the centroid divides a median in a 2:1 ratio, the longer part (XG) is 8 units, and the shorter part (GP) is 4 units.

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11) The centroid always lies:

Explanation

Unlike other centers (like the orthocenter or circumcenter), the centroid is always inside the triangle, no matter the shape.

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12) The centroid of a triangle with vertices (2,4), (8,1), and (5,7) is:

Explanation

Average the coordinates:

x = (2 + 8 + 5)/3 = 5

y = (4 + 1 + 7)/3 = 4 → (5,4).

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13) The centroid is the point of concurrency of:

Explanation

The three medians of a triangle always intersect at the centroid.

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14) In △XYZ, the centroid divides median XP into two segments. If XP= 18, then the distance from X to the centroid is:

Explanation

The centroid divides the median in a 2:1 ratio, so:

2/3 × 18 = 12 units from the vertex to the centroid.

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15) In an isosceles triangle, the centroid lies on the:

Explanation

In an isosceles triangle, the centroid lies along the altitude, angle bisector, and perpendicular bisector from the vertex — all these lines overlap.

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16) Which is NOT true about medians?

Explanation

Medians connect vertices to midpoints, but they are not necessarily perpendicular. Only altitudes are perpendicular.

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17) A cardboard triangle is balanced on the tip of a pencil. The balance point represents the triangle's:

Explanation

The centroid is the balance point — the point where the entire triangle’s weight would balance evenly.

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18) Which property of triangles can be proven using medians?

Explanation

This is the centroid theorem, proven by geometry or coordinate reasoning — the medians always divide in a 2:1 ratio.

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19) A triangle has vertices (2,2), (8,2), and (2,8). Its centroid is:

Explanation

The centroid of a triangle is found by averaging the x-coordinates and y-coordinates of its vertices. The x-coordinates of the vertices (2, 2) and (8, 2) average to (2 + 8)/3 = 4, and the y-coordinates (2, 8) average to (2 + 8)/3 = 4. Hence, the centroid is (4,4).

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20) The centroid theorem is sometimes called the:

Explanation

The theorem stating all medians meet at one point is known as the Concurrency of Medians Theorem.

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Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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If all three medians are drawn in a triangle, how many smaller...
The centroid divides a median so that the part closer to the vertex...
In △PQR, the medians divide the triangle into six regions of equal...
A median of a triangle is drawn from a vertex to ____.
In △ABC, medians AD, BE, and CF intersect at centroid G. Which of...
Which point of concurrency is the intersection of the medians?
The centroid divides each median in the ratio ____.
If a median is 15 units long, the distance from the centroid to the...
The centroid is also called the ____.
In △XYZ, medians XP, YQ, and ZR intersect at centroid G. If XP=12,...
The centroid always lies:
The centroid of a triangle with vertices (2,4), (8,1), and (5,7) is:
The centroid is the point of concurrency of:
In △XYZ, the centroid divides median XP into two segments. If XP=...
In an isosceles triangle, the centroid lies on the:
Which is NOT true about medians?
A cardboard triangle is balanced on the tip of a pencil. The balance...
Which property of triangles can be proven using medians?
A triangle has vertices (2,2), (8,2), and (2,8). Its centroid is:
The centroid theorem is sometimes called the:
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