Medians and Centroid Proof Applications

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.

The centroid theorem states that the medians of a triangle intersect at the centroid, which divides each median into a 2:1 ratio and results in the division of the triangle into six smaller triangles of equal area.

Explanation

The centroid theorem states that the medians of a triangle intersect at the centroid, which divides each median into a 2:1 ratio and results in the division of the triangle into six smaller triangles of 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

A median connects a vertex of a triangle to the midpoint of the opposite side, effectively dividing the triangle into two smaller triangles of equal area.

Explanation

A median connects a vertex of a triangle to the midpoint of the opposite side, effectively dividing the triangle into two smaller triangles of equal area.

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 the three medians of a triangle intersect. It is also the center of mass or balance point of the triangle.

Explanation

The centroid is the point where the three medians of a triangle intersect. It is also the center of mass or balance point of the triangle.

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

1:1

2:1

3:1

1:2

The centroid of a triangle divides each median into two segments, with the segment connecting the centroid to the vertex being twice the length of the segment connecting the centroid to the midpoint of the opposite side, thus the ratio is 2:1.

Explanation

The centroid of a triangle divides each median into two segments, with the segment connecting the centroid to the vertex being twice the length of the segment connecting the centroid to the midpoint of the opposite side, thus the ratio is 2:1.

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

In a triangle, the centroid divides each median into two segments, one of which is twice as long as the other. Thus, the distance from the centroid to the midpoint of a side is one-third of the length of the median. Therefore, for a median of 15 units, the distance is 15/3 = 5 units.

Explanation

In a triangle, the centroid divides each median into two segments, one of which is twice as long as the other. Thus, the distance from the centroid to the midpoint of a side is one-third of the length of the median. Therefore, for a median of 15 units, the distance is 15/3 = 5 units.

7) The centroid is also called the ____.

Angle bisector

Balancing point

Perpendicular bisector

Orthocenter

The centroid of a triangle is the point at which the three medians intersect, and it is often referred to as the 'balancing point' because it is the point that would balance the triangle if it were made of uniform material.

Explanation

The centroid of a triangle is the point at which the three medians intersect, and it is often referred to as the 'balancing point' because it is the point that would balance the triangle if it were made of uniform material.

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

In a triangle, the centroid divides each median into a ratio of 2:1. Since median XP = 12, the lengths are divided such that XG is 8 (2 parts) and GP is 4 (1 part).

Explanation

In a triangle, the centroid divides each median into a ratio of 2:1. Since median XP = 12, the lengths are divided such that XG is 8 (2 parts) and GP is 4 (1 part).

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

2

3

4

6

When all three medians of a triangle are drawn, they intersect at a single point called the centroid. This point divides each median into two segments with a 2:1 ratio. The three medians divide the triangle into six smaller triangles, all of which have equal area, as they share a common vertex at the centroid and the base on the triangle's sides.

Explanation

When all three medians of a triangle are drawn, they intersect at a single point called the centroid. This point divides each median into two segments with a 2:1 ratio. The three medians divide the triangle into six smaller triangles, all of which have equal area, as they share a common vertex at the centroid and the base on the triangle's sides.

10) The centroid always lies:

On the outside

At a vertex

Inside the triangle

On the longest side

The centroid is the point where the three medians of a triangle intersect, and it is always located inside the triangle, regardless of the triangle's type.

Explanation

The centroid is the point where the three medians of a triangle intersect, and it is always located inside the triangle, regardless of the triangle's type.

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

(5,4)

(3,4)

(5,5)

(4,4)

The centroid of a triangle is calculated by taking the average of the x-coordinates and the average of the y-coordinates of the vertices. For the given vertices, the x-coordinates are 2, 8, and 5, and the y-coordinates are 4, 1, and 7. The centroid is therefore ((2 + 8 + 5)/3, (4 + 1 + 7)/3) = (5, 4).

Explanation

The centroid of a triangle is calculated by taking the average of the x-coordinates and the average of the y-coordinates of the vertices. For the given vertices, the x-coordinates are 2, 8, and 5, and the y-coordinates are 4, 1, and 7. The centroid is therefore ((2 + 8 + 5)/3, (4 + 1 + 7)/3) = (5, 4).

12) The centroid is the point of concurrency of:

Medians

Altitudes

Angle bisectors

Perpendicular bisectors

The centroid of a triangle is the point where the three medians intersect. A median connects a vertex of the triangle to the midpoint of the opposite side, and the centroid divides each median into a ratio of 2:1.

Explanation

The centroid of a triangle is the point where the three medians intersect. A median connects a vertex of the triangle to the midpoint of the opposite side, and the centroid divides each median into a ratio of 2:1.

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 of a triangle divides each median into two segments with a ratio of 2:1. Therefore, if the total length of median XP is 18, the distance from point X to the centroid is 12 (which is 2/3 of the median length).

Explanation

The centroid of a triangle divides each median into two segments with a ratio of 2:1. Therefore, if the total length of median XP is 18, the distance from point X to the centroid is 12 (which is 2/3 of the median length).

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, which is the point of intersection of the medians, lies along the angle bisector, perpendicular bisector, and altitude from the vertex angle due to the triangle's symmetrical properties.

Explanation

In an isosceles triangle, the centroid, which is the point of intersection of the medians, lies along the angle bisector, perpendicular bisector, and altitude from the vertex angle due to the triangle's symmetrical properties.

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 a triangle, the centroid divides each median into two segments, with the segment closer to the vertex being twice as long as the segment closer to the midpoint of the opposite side.

Explanation

In a triangle, the centroid divides each median into two segments, with the segment closer to the vertex being twice as long as the segment closer to the midpoint of the opposite side.

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 are line segments from each vertex to the midpoint of the opposite side and are not necessarily perpendicular bisectors of the sides. While every triangle does have three medians and they meet at the centroid, they do not have to be perpendicular to the sides they bisect.

Explanation

Medians are line segments from each vertex to the midpoint of the opposite side and are not necessarily perpendicular bisectors of the sides. While every triangle does have three medians and they meet at the centroid, they do not have to be perpendicular to the sides they bisect.

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 of a triangle is the point where all three medians intersect and is the balance point of the triangle, allowing it to be balanced evenly.

Explanation

The centroid of a triangle is the point where all three medians intersect and is the balance point of the triangle, allowing it to be balanced evenly.

18) The centroid theorem is sometimes called the:

Concurrency of Perpendicular Bisectors

Concurrency of Angle Bisectors

Concurrency of Medians

Concurrency of Altitudes

The centroid theorem states that the medians of a triangle intersect at a single point, known as the centroid, which is the center of mass of the triangle. This is why it is referred to as the concurrency of Medians.

Explanation

The centroid theorem states that the medians of a triangle intersect at a single point, known as the centroid, which is the center of mass of the triangle. This is why it is referred to as the concurrency of Medians.

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

The centroid of a triangle is the point where all three medians intersect, and it divides each median into two segments with a ratio of 2:1, meaning that the segment connecting the centroid to a vertex is twice as long as the segment connecting the centroid to the midpoint of the opposite side.

Explanation

The centroid of a triangle is the point where all three medians intersect, and it divides each median into two segments with a ratio of 2:1, meaning that the segment connecting the centroid to a vertex is twice as long as the segment connecting the centroid to the midpoint of the opposite side.

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).