Medians and Altitudes — Coordinate Geometry Applications

From vertex to centroid is twice the distance from centroid to midpoint, giving ratio 2:1. The centroid lies two-thirds of the way from each vertex to the opposite midpoint. Option A inverts the ratio. Options C and D have no basis in the centroid's definition.

The centroid lies along every median exactly two-thirds of the way from each vertex, dividing the median in a 2:1 ratio from vertex to midpoint. Option A incorrectly describes the centroid as related to altitudes — it is defined by medians. Option C is false since the centroid always lies inside any triangle. Option D describes the incenter not the centroid.

The answer is True. The centroid lies exactly two-thirds of the way from each vertex to the midpoint of the opposite side, creating a 2:1 ratio in every median. This is consistent across all triangles regardless of shape and can be verified both geometrically and through coordinate analysis.

x = (-3+6+3)/3 = 6/3 = 2. y = (4+1+7)/3 = 12/3 = 4. Centroid = (2,4). Option A gives (2,5) with wrong y. Option B gives (3,4) with wrong x. Option D gives (1,3) with both coordinates wrong.

The slope of BC is 2. The altitude is perpendicular to BC so its slope is -1/2, the negative reciprocal of 2. Check: 2 × (-1/2) = -1 confirming perpendicularity. Option A gives the slope of BC itself. Option C gives -2, which is the negative reciprocal of 1/2 not 2. Option D gives 1/2.

The slope of the given line is 1/2. The perpendicular slope is the negative reciprocal: -1/(1/2) = -2. The perpendicular line has equation y = -2x + k where k depends on the specific point it passes through. Option A uses the same slope. Option B uses +2 not -2. Option D uses -1/2 which is not the negative reciprocal of 1/2.

Using the 2:1 ratio from vertex through centroid to midpoint: vertex x = 3 + 2(5-3) = 3+4 = 7. Vertex y = 4 + 2(7-4) = 4+6 = 10. Vertex = (7,10). Option A gives (6,9), option C gives (8,11), option D gives (9,12), none of which correctly apply the section ratio.

An altitude forms a right angle with the opposite side by definition. This perpendicularity distinguishes altitudes from medians which connect to midpoints and from angle bisectors which divide angles. The altitude may meet the extension of BC in obtuse triangles but always remains perpendicular to it.

The answer is False. The altitude passes through the orthocenter, not the centroid. These are different points except in equilateral triangles where all centers coincide. The centroid is defined by medians, not altitudes, so altitudes generally miss the centroid entirely.

x = (2+6+10)/3 = 18/3 = 6. y = (4+8+4)/3 = 16/3 ≈ 5.33. Centroid = (6, 5.33). Option B gives (5,6) with wrong averages. Option C gives (7,5). Option D gives (4,5). Only option A correctly averages all three coordinates.

Average the coordinates: x = (2+6)/2 = 4 and y = (4+8)/2 = 6, giving midpoint (4,6). Option A gives (3,5) using wrong averages. Option B gives (5,7), also incorrect. Option C gives (2,8), taking one coordinate from each point rather than averaging.

Slope = (4-0)/(3-0) = 4/3. Through the origin the equation is y = (4/3)x. Option A inverts the slope giving 3/4. Option C gives slope 2, which does not pass through (3,4) from the origin. Option D gives slope 1, also incorrect.

Slope of BC = (4-2)/(5-1) = 2/4 = 1/2. The altitude is perpendicular to BC so its slope is the negative reciprocal: -1/(1/2) = -2. Check: (1/2)×(-2) = -1 confirming perpendicularity. Option B gives the slope of BC itself. Options C and D are incorrect.

The centroid coordinates are the arithmetic mean of all three vertex coordinates in each direction. Dividing by 3 ensures the result is the average of the three values. Option A uses only two vertices and divides by 2. Option B sums without dividing. Option D divides by 2 instead of 3.

Midpoint M of BC = ((6+0)/2, (0+8)/2) = (3,4). Length AM = √((3-0)²+(4-0)²) = √(9+16) = √25 = 5. This is a 3-4-5 right triangle relationship. Options B, C, D all give lengths that do not result from this distance calculation.

The answer is True. A median is defined as the segment connecting a vertex to the midpoint of the opposite side. By definition of midpoint, the opposite side is bisected into two segments of equal length. This is what distinguishes a median from an altitude or angle bisector.

Midpoint M of BC = ((6+10)/2, (8+4)/2) = (8,6). Slope AM = (6-4)/(8-2) = 2/6 = 1/3. Equation: y - 4 = (1/3)(x - 2). Option A uses slope 1 instead of 1/3. Option C gives a different line entirely. Option D uses wrong slope and point.

The answer is False. The orthocenter and centroid coincide only in equilateral triangles where all centers merge due to perfect symmetry. In scalene or isosceles triangles these are distinct points with different geometric definitions and different coordinate positions.

x = (0+6+3)/3 = 9/3 = 3. y = (0+0+6)/3 = 6/3 = 2. Centroid = (3,2). Option A gives (2,3) swapping x and y. Option B gives (3,3) with wrong y. Option D gives (4,2) with wrong x.

The answer is False. The centroid is the average of the three vertex coordinates, which always places it inside the convex region bounded by the triangle. Unlike the orthocenter which can lie outside for obtuse triangles, the centroid is always interior regardless of the triangle's shape.