Medians and Altitudes — Conceptual & Computational Understanding

In a right triangle each leg is already perpendicular to the other, so the altitudes from the two acute vertices coincide with the legs and both pass through the right-angle vertex. This makes the right-angle vertex itself the orthocenter. Options C and D describe locations that do not correspond to any triangle center.

The centroid equals the average of the three vertex coordinates and always lies inside the triangle regardless of its shape. It represents the triangle's center of mass or balance point. Option A is where the orthocenter of an obtuse triangle lies. Option B describes no standard triangle center. Option C is where the circumcenter lies relative to the circumscribed circle.

Every triangle has exactly three medians and three altitudes, one from each vertex. Medians connect vertices to midpoints while altitudes are perpendicular to the opposite side. Option A describes only altitudes. Option C is false since every triangle has three medians. Option D is false since median and altitude coincide only in special cases like equilateral triangles.

All three medians are concurrent — they intersect at exactly one point called the centroid. This is guaranteed by classical Euclidean geometry for any triangle. Option A is false since medians converge not run parallel. Option B describes altitudes not medians. Option D contradicts the definition of concurrency.

Centroid = ((2+8+4)/3, (2+2+6)/3) = (14/3, 10/3) ≈ (4.67, 3.33). Option A gives (4,3) with wrong x average. Option B gives (5,3) with both averages wrong. Option C gives correct y but wrong x coordinate.

The answer is False. Altitudes lie inside only for acute triangles. For obtuse triangles the altitudes from the acute vertices must be extended outside the triangle to meet the opposite side's extension, placing the orthocenter outside. In a right triangle one altitude coincides with each leg.

Centroid = ((0+6+3)/3, (0+0+6)/3) = (9/3, 6/3) = (3, 2). Option A gives (2,3) swapping x and y. Option B gives (3,3) using wrong y average. Option D gives (2,2) with wrong calculations for both coordinates.

An altitude always forms a right angle with the side it meets or its extension, making it perpendicular by definition. This perpendicularity distinguishes altitudes from medians and angle bisectors. Option A gives parallel which is the opposite relationship. Options B and D describe different geometric properties.

Median = (√3/2) times side = (√3/2) times 12 = 6√3 cm. In an equilateral triangle the median is also the altitude, perpendicular bisector, and angle bisector. Option A gives 8, option B gives 4√3, option D gives 12, none of which correctly apply the median formula.

The orthocenter is defined as the common intersection point of all three altitudes of a triangle. The centroid is where medians meet. The circumcenter is where perpendicular bisectors of the sides meet. The incenter is where the angle bisectors meet.

The three medians of any triangle intersect at the centroid, which always lies inside the triangle. It divides every median in a 2:1 ratio from vertex to midpoint and represents the triangle's center of mass. The orthocenter is where altitudes meet. The incenter is where angle bisectors meet. The circumcenter is where perpendicular bisectors meet.

From vertex to centroid is always twice as long as from centroid to midpoint, giving the universal ratio of 2:1. This means the centroid is located two-thirds of the total median length from the vertex. Option A gives equal halves. Option B gives 3:2. Option C inverts the correct ratio.

The centroid divides the median in a 2:1 ratio from vertex to midpoint. So GD = AG divided by 2 = 10 divided by 2 = 5 cm. Option A gives 3, option C gives 7, option D gives 8, none of which correctly apply the 2:1 ratio.

The answer is False. The centroid divides each median into unequal segments in the fixed ratio of 2:1. The longer portion always lies between the vertex and the centroid, and the shorter portion lies between the centroid and the midpoint of the opposite side.

The centroid lies two-thirds of the way from any vertex along the median to the midpoint. This means AG is twice GD, giving the ratio 2:1 from vertex to midpoint. Option A inverts the ratio. Options C and D are incorrect ratios that do not apply to the centroid.

The answer is False. Only altitudes are defined as perpendicular to their opposite sides. Medians simply connect each vertex to the midpoint of the opposite side and form right angles with the opposite side only in special cases such as isosceles triangles where the median to the base is also the altitude.

All three altitudes, which are perpendicular segments from each vertex to the opposite side, intersect at the orthocenter. Its position varies — inside for acute triangles, at the right-angle vertex for right triangles, and outside for obtuse triangles. The centroid is where medians meet. The incenter is where angle bisectors meet.

The answer is False. In an obtuse triangle the altitudes from the two acute vertices fall outside the triangle's boundaries, causing their intersection point — the orthocenter — to lie outside the triangle. Only in acute triangles does the orthocenter lie inside.

A median always joins a vertex to the midpoint of the opposite side, ensuring the median divides the entire triangle into two regions of equal area. Option A describes an endpoint which is a vertex not a midpoint. Option C describes where an altitude meets a side. Option D describes a point on the angle bisector.

The answer is True. In an equilateral triangle all four classical centers — centroid, orthocenter, circumcenter, and incenter — coincide at the same point due to the triangle's complete symmetry in angles, sides, and altitudes. This is a unique property of equilateral triangles that does not hold for other triangle types.