Centroid Theorems & Median Proofs

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

The midpoint theorem.

The centroid theorem.

The triangle sum theorem.

The Pythagorean theorem.

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

Explanation

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

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

The midpoint of the opposite side

The incenter

The circumcenter

The orthocenter

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

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.

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

Each median is perpendicular to the opposite side.

The centroid divides each median into two equal parts.

The centroid divides each median so that AG:GD = 2:1.

The centroid lies on the circumcircle of △ABC.

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.

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.

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

Centroid

Orthocenter

Incenter

Circumcenter

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

Explanation

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

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

1:1

2:1

3:1

1:2

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.

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.

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

10 units

7:5 units

5 units

12 units

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

15 ÷ 3 = 5 units.

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.

7) The centroid is also called the ____.

Angle bisector

Balancing point

Perpendicular bisector

Orthocenter

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.

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.

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

XG=6, GP=6

XG=8, GP=4

XG=4, GP=8

XG=9, GP=3

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.

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.

9) If all three medians are drawn in a triangle, how many smaller triangles of equal area are formed?

2

3

4

6

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

Explanation

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

10) The centroid always lies:

On the outside

At a vertex

Inside the triangle

On the longest side

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

Explanation

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

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

(5,4)

(3,4)

(5,5)

(4,4)

Average the coordinates:

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

y = (4 + 1 + 7)/3 = 4 → (5,4).

Explanation

Average the coordinates:

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

y = (4 + 1 + 7)/3 = 4 → (5,4).

12) The centroid is the point of concurrency of:

Medians

Altitudes

Angle bisectors

Perpendicular bisectors

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

Explanation

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

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

6

9

12

15

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

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

Explanation

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

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

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

Angle bisector of the vertex angle

Perpendicular bisector of the base

Altitude from the vertex angle

All of the above

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

Explanation

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

15) The centroid divides a median so that the part closer to the vertex is:

Equal to the other part

Twice as long as the other part

Shorter than the other part

Unrelated to the other part

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

Explanation

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

16) Which is NOT true about medians?

Every triangle has 3 medians

They always intersect inside the triangle

They are always perpendicular bisectors

They are concurrent at the centroid

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

Explanation

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

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

Incenter

Circumcenter

Orthocenter

Centroid

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

Explanation

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

18) The centroid theorem is sometimes called the:

Concurrency of Perpendicular Bisectors

Concurrency of Angle Bisectors

Concurrency of Medians

Concurrency of Altitudes

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

Explanation

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

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

The centroid divides each median in a 2:1 ratio

The centroid lies on the circumscribed circle

The centroid is equidistant from each vertex

The centroid bisects each angle

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

Explanation

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

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

(4,4)

(6,4)

(4,6)

(2,2)

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).

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).