Medians and Altitudes — Conceptual & Computational Understanding

  • 9th Grade
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| Questions: 20 | Updated: Dec 17, 2025
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1) The point where all medians of a triangle meet is called the

Explanation

The three medians of any triangle intersect at the centroid, a special balance point that always lies inside the triangle and divides every median into a 2:1 ratio measured from the vertex toward the midpoint.
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About This Quiz
Medians And Altitudes Conceptual & Computational Understanding - Quiz

How do medians and altitudes reveal structure inside a triangle? In this quiz, you’ll explore both concepts through diagrams, relationships, and guided reasoning. You’ll practice identifying each segment, work through problems that connect geometry with algebraic relationships, and see how these lines relate to balance, symmetry, and distance. Step by... see morestep, you’ll deepen both your intuition and your computational accuracy.
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2) A median of a triangle connects a vertex to the _____ of the opposite side.

Explanation

A median always joins a vertex to the midpoint of the opposite side, ensuring that each median divides the entire triangle into two regions of equal area.
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3) The altitudes of a triangle are concurrent at the

Explanation

All three altitudes, which are perpendicular segments from a vertex to the opposite side, intersect at the orthocenter, a point whose location depends on whether the triangle is acute, right, or obtuse.

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4) In ΔABC, the centroid G divides median AD in the ratio AG : GD = 2 : 1 (from vertex to midpoint).

Explanation

Because the centroid lies two-thirds of the way from a vertex along any median, AG is twice GD, giving the standard centroid ratio of 2:1 from vertex to midpoint.

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5) In ΔABC, if AD is a median and AG = 10 cm, find GD.

Explanation

Since the centroid cuts the median in a 2:1 ratio with the longer portion from the vertex, GD must be half of AG, giving 5 cm as the remaining segment.
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6) In a right triangle, the orthocenter lies

Explanation

 For a right triangle, each altitude from the legs meets at the right-angle vertex itself, making that vertex the orthocenter because the perpendicular directions already exist in the triangle’s structure.

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7) In an equilateral triangle of side 12 cm, find the length of the median.

Explanation

Using the formula median = (√3/2)·side, substituting 12 gives 6√3 cm, and in equilateral triangles this median is also the altitude, perpendicular bisector, and angle bisector all at once.

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8) The centroid of a triangle always lies

Explanation

No matter the triangle’s shape, its centroid—being the average of the three vertex coordinates—always lies inside the triangle because it represents the center of mass.
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9) In ΔPQR with coordinates P(0, 0), Q(6, 0), R(3, 6), the centroid is

Explanation

Applying the centroid formula ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) gives coordinates (3, 2), reflecting the triangle’s balance point in the coordinate plane.
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10) Which statement is true about medians and altitudes?

Explanation

Every triangle has exactly three medians and three altitudes, one from each vertex, because medians depend on midpoints while altitudes depend on perpendicularity, and both exist for all triangles.

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11) The centroid divides each median in the ratio _____ (from vertex to midpoint).

Explanation

From vertex to centroid is twice as long as centroid to midpoint, establishing the universal centroid ratio of 2:1 along each median.
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12) The point of intersection of the three altitudes is called the _____.

Explanation

The orthocenter is defined as the common intersection point of the three altitudes of a triangle, regardless of where this point lies relative to the triangle’s interior or exterior.
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13) In a triangle, an altitude is always _____ to the side it meets.

Explanation

An altitude always forms a right angle with the side it touches (or its extension), making it the perpendicular height from the vertex.

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14) In a triangle with vertices A(2, 2), B(8, 2), C(4, 6), the centroid is at _____.

Explanation

Computing averages of the x-coordinates and y-coordinates gives (4.67, 3.33), meaning the centroid is located at the arithmetic center of the triangle’s vertices.
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15) The medians of a triangle are _____ lines.

Explanation

All three medians are concurrent because they intersect at the centroid, a single point guaranteed by classical Euclidean geometry.
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16) The centroid and orthocenter of an equilateral triangle are the same point.

Explanation

This is true because in an equilateral triangle all centers—centroid, orthocenter, circumcenter, and incenter—coincide due to complete symmetry in angles, sides, and altitudes.

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17) The orthocenter of an obtuse triangle lies inside the triangle.

Explanation

This is false because in an obtuse triangle the altitudes from the acute vertices fall outside the triangle’s boundaries, causing the orthocenter to lie outside the triangle.
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18) In any triangle, the medians are perpendicular to the sides.

Explanation

This is false because only altitudes are defined as perpendicular, while medians simply connect vertices to midpoints and rarely form right angles.
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19) The centroid divides each median into two equal halves.

Explanation

This is false since the centroid divides each median into unequal segments with a fixed ratio of 2:1, with the longer portion always adjacent to the vertex.
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20) The altitudes of a triangle are always inside the triangle.

Explanation

This is false because altitudes are inside an acute triangle but extend outside when the triangle is obtuse, and only in a right triangle does one altitude coincide with a side.

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The point where all medians of a triangle meet is called the
A median of a triangle connects a vertex to the _____ of the opposite...
The altitudes of a triangle are concurrent at the
In ΔABC, the centroid G divides median AD in the ratio AG : GD =...
In ΔABC, if AD is a median and AG = 10 cm, find GD.
In a right triangle, the orthocenter lies
In an equilateral triangle of side 12 cm, find the length of the...
The centroid of a triangle always lies
In ΔPQR with coordinates P(0, 0), Q(6, 0), R(3, 6), the centroid...
Which statement is true about medians and altitudes?
The centroid divides each median in the ratio _____ (from vertex to...
The point of intersection of the three altitudes is called the _____.
In a triangle, an altitude is always _____ to the side it meets.
In a triangle with vertices A(2, 2), B(8, 2), C(4, 6), the centroid is...
The medians of a triangle are _____ lines.
The centroid and orthocenter of an equilateral triangle are the same...
The orthocenter of an obtuse triangle lies inside the triangle.
In any triangle, the medians are perpendicular to the sides.
The centroid divides each median into two equal halves.
The altitudes of a triangle are always inside the triangle.
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