Medians and the Centroid Basics

Grade 10th

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| By Thames

Thames

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| Questions: 20

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1) A median of a triangle is a segment that connects a vertex to:

The opposite side’s midpoint

The opposite side’s endpoint

The orthocenter

The altitude’s foot

A median is defined as a line segment that connects a vertex of a triangle to the midpoint of the opposite side, thereby dividing that side into two equal lengths.

Explanation

A median is defined as a line segment that connects a vertex of a triangle to the midpoint of the opposite side, thereby dividing that side into two equal lengths.

Submit

About This Quiz

Are you ready to learn about the medians of a triangle? In this quiz, you’ll discover how medians connect a vertex to the midpoint of the opposite side and always meet at one special point — the centroid. You’ll practice finding midpoints, working with coordinates, and learning how the centroid... see moresplits each median into two parts. Step by step, you’ll see how the centroid is like the triangle’s balance point! see less

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2) In triangle ABC, point D is the midpoint of BC. Which segment is the median from vertex A?

In a triangle, a median is a line segment that connects a vertex to the midpoint of the opposite side. Since D is the midpoint of BC, the segment AD represents the median from vertex A.

Explanation

In a triangle, a median is a line segment that connects a vertex to the midpoint of the opposite side. Since D is the midpoint of BC, the segment AD represents the median from vertex A.

Submit

3) A median of a triangle is a segment that connects a vertex to:

The opposite side’s midpoint

The opposite side’s endpoint

The orthocenter

The altitude’s foot

In geometry, the median of a triangle is defined as the line segment that connects a vertex to the midpoint of the opposite side, thus dividing the triangle into two smaller triangles of equal area.

Explanation

In geometry, the median of a triangle is defined as the line segment that connects a vertex to the midpoint of the opposite side, thus dividing the triangle into two smaller triangles of equal area.

Submit

4) A median of a triangle is a segment that connects a vertex to:

The opposite side’s midpoint

The opposite side’s endpoint

The orthocenter

The altitude’s foot

A median is defined as a line segment that connects a vertex of a triangle to the midpoint of the opposite side, thereby dividing that side into two equal lengths.

Explanation

A median is defined as a line segment that connects a vertex of a triangle to the midpoint of the opposite side, thereby dividing that side into two equal lengths.

Submit

5) In triangle ABC, point D is the midpoint of BC. Which segment is the median from vertex A?

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Since D is the midpoint of BC, the median from vertex A is segment AD.

Explanation

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Since D is the midpoint of BC, the median from vertex A is segment AD.

Submit

6) If the coordinates of the vertices of a triangle are (0,0), (6,0), and (0,8), what are the coordinates of the centroid?

(2,2)

(2, 8/3)

(3,4)

(2, 8)

The centroid of a triangle can be found by averaging the x-coordinates and the y-coordinates of its vertices. Here, the x-coordinates are 0, 6, and 0, and the y-coordinates are 0, 0, and 8. Thus, the centroid is ((0+6+0)/3, (0+0+8)/3) = (2, 8/3).

Explanation

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7) Which theorem explains why alternate interior angles are congruent when a transversal intersects two parallel lines?

Triangle Inequality Theorem

Corresponding Angles Postulate

Alternate Interior Angles Theorem

Pythagorean Theorem

The Alternate Interior Angles Theorem states that when a transversal crosses two parallel lines, the pairs of alternate interior angles are equal in measure. This theorem is fundamental in understanding the properties of angles formed by parallel lines and transversals.

Explanation

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8) The centroid coordinates of a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) are:

((x₁+x₂)/2, (y₁+y₂)/2)

((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)

((x₁+x₂−x₃)/3, (y₁+y₂−y₃)/3)

((x₁+x₂+x₃)/2, (y₁+y₂+y₃)/2)

The centroid of a triangle, often referred to as the geometric center, is calculated by taking the average of the x-coordinates and the average of the y-coordinates of the triangle's vertices. Therefore, the correct formula is the sum of all x-coordinates divided by 3 and the sum of all y-coordinates divided by 3.

Explanation

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9) The centroid divides each median so that the distance from the centroid to the midpoint of a side is:

Twice the distance from the centroid to the opposite vertex:

Equal to the distance from the centroid to the opposite vertex:

One-third the median length:

Two-thirds the median length:

The centroid of a triangle divides each median in a 2:1 ratio, where the longer segment is between the centroid and the vertex, and the shorter segment is between the centroid and the midpoint of the opposite side. Therefore, the distance from the centroid to the midpoint is two-thirds of the median's length.

Explanation

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10) If the median from vertex A of triangle ABC is 12 units long, how far is the centroid from vertex A?

3 units

4 units

6 units

8 units

The centroid of a triangle is located two-thirds of the way along the median from the vertex. Therefore, if the median is 12 units long, the distance from vertex A to the centroid would be 12 * (2/3) = 8 units.

Explanation

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11) A median of a triangle is a segment that connects a vertex to:

The opposite side’s midpoint

The opposite side’s endpoint

The orthocenter

The altitude’s foot

In geometry, the median of a triangle is defined as the line segment that connects a vertex to the midpoint of the opposite side, thus dividing the triangle into two smaller triangles of equal area.

Explanation

In geometry, the median of a triangle is defined as the line segment that connects a vertex to the midpoint of the opposite side, thus dividing the triangle into two smaller triangles of equal area.

Submit

12) In triangle ABC, point D is the midpoint of BC. Which segment is the median from vertex A?

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Since D is the midpoint of BC, the median from vertex A is segment AD.

Explanation

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Since D is the midpoint of BC, the median from vertex A is segment AD.

Submit

13) If the median from vertex A of triangle ABC is 12 units long, how far is the centroid from vertex A?

3 units

4 units

6 units

8 units

The centroid of a triangle divides each median into two segments, one of which is twice as long as the other. Since the median is 12 units, the distance from vertex A to the centroid is 8 units, which is one-third of the median length.

Explanation

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14) The centroid divides each median into two segments. Which is true about these segments?

The shorter segment is twice the longer:

They vary depending on the triangle:

They are equal:

The longer segment is twice the shorter:

In a triangle, the centroid divides each median into two segments in a 2:1 ratio, where the longer segment (from the vertex to the centroid) is twice the length of the shorter segment (from the centroid to the midpoint of the opposite side).

Explanation

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15) A triangle has medians of lengths 5, 6, and 7. Which of the following is true about the medians?

They may not intersect at a single point:

They form a triangle with sides equal to the lengths of the medians:

They are concurrent at the orthocenter:

They are concurrent at the centroid:

In any triangle, the three medians always intersect at a single point called the centroid, which divides each median into two segments with a 2:1 ratio.

Explanation

In any triangle, the three medians always intersect at a single point called the centroid, which divides each median into two segments with a 2:1 ratio.

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16) The centroid of a triangle always lies:

Inside the triangle

On a vertex

Outside 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 type of triangle.

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 type of triangle.

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17) In a right triangle, the centroid lies:

At the right angle vertex

On the hypotenuse

Inside the triangle

At the midpoint of the hypotenuse

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

Explanation

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

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18) In an equilateral triangle, the centroid is also the:

Incenter

Circumcenter

Orthocenter

All of the above

In an equilateral triangle, the centroid, incenter, circumcenter, and orthocenter all coincide at the same point due to the symmetry of the triangle. This means that they all represent the same point in the case of an equilateral triangle.

Explanation

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19) Which statement is false?

Every triangle has exactly 3 medians

The medians are concurrent at the centroid

The centroid is always equidistant from the vertices

The centroid is inside the triangle

The centroid is the point where the three medians intersect, but it is not equidistant from the vertices of the triangle. Its distance from each vertex varies based on the shape of the triangle.

Explanation

The centroid is the point where the three medians intersect, but it is not equidistant from the vertices of the triangle. Its distance from each vertex varies based on the shape of the triangle.

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20) Which theorem could you use in a coordinate proof to show that the medians of a triangle are concurrent at the centroid?

Pythagorean Theorem

Law of Cosines

Midpoint Theorem

Sine Rule

The Midpoint Theorem is used in coordinate proofs to show that the medians of a triangle intersect at a single point, known as the centroid. This is because each median connects a vertex to the midpoint of the opposite side, creating concurrent lines.

Explanation

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