How to Find the Centroid When the Right Angle Is Not at the Origin

1.The shortcut is relative to the right-angle vertex

Place a right triangle with the right angle at the origin and the legs along the axes: vertices (0, 0), (b, 0), (0, h). Then G = (b/3, h/3) from that vertex. That is the same statement as in How to Find the Centroid of a Right Triangle at (b/3, h/3).

“From the right-angle vertex” is a local frame. Sliding the triangle across the plane does not freeze G at (b/3, h/3) in absolute x and y. The centroid rides with the triangle. You either carry the offset of the right-angle vertex, or you forget the shortcut and average the three given points. The same frame choice, without a translated homework triangle, is How to Measure the Centroid from the Right-Angle Vertex.

\[ G_{\text{origin}}=\Bigl(\tfrac{b}{3},\tfrac{h}{3}\Bigr)\quad\text{(right angle at }(0,0)\text{)} \]

2.The common mistake: (b/3, h/3) as absolute coordinates

A typical prompt gives A(3, 1), B(9, 1), C(3, 5). The right angle is at A, the legs are still 6 and 4, so someone writes G = (6/3, 4/3) = (2, 4/3) and plots that pair on the same axes as A, B, and C. That point is the centroid of a copy sitting at the origin. It is not the centroid of the triangle you were given: x = 2 lies left of the whole figure (x runs from 3 to 9).

The mistake is treating a relative offset as a location. (b/3, h/3) answers “how far from the right angle along the legs,” not “what are the worksheet coordinates?” If your G falls outside a clearly interior right triangle, you almost certainly skipped the translation.

\[ (2,\tfrac{4}{3})\neq G\text{ when the right angle is at }(3,1) \]

3.Method 1: translate from the right-angle vertex

Let the right-angle vertex be (xᵣ, yᵣ). If the legs are parallel to the axes, signed lengths b (horizontal) and h (vertical) point along those legs — positive right and up, negative left and down. Then G = (xᵣ + b/3, yᵣ + h/3). You are adding the origin-frame centroid as a vector.

Signs matter. Right angle at (5, 2), other vertices (11, 2) and (5, −2): b = +6 and h = −4, so G = (5 + 2, 2 − 4/3) = (7, 2/3). Dropping the minus and writing (7, 2 + 4/3) puts G above the base instead of below it.

\[ G=\bigl(x_r+\tfrac{b}{3},\; y_r+\tfrac{h}{3}\bigr) \]

4.Worked example: right angle at (3, 1), legs 6 and 4

Vertices A(3, 1), B(9, 1), C(3, 5). Horizontal leg b = 9 − 3 = 6, vertical leg h = 5 − 1 = 4, right angle at A = (3, 1). Translate: G = (3 + 6/3, 1 + 4/3) = (3 + 2, 1 + 4/3) = (5, 7/3) ≈ (5, 2.33).

The faint triangle in the figure is the origin copy whose centroid is (2, 4/3) — the pair you would plot if you applied the shortcut as absolute coordinates. The solid triangle is the given one; its centroid is the filled G at (5, 7/3). The coral arrow is the translation (3, 1) from the origin to the actual right angle.

\[ G=\bigl(3+\tfrac{6}{3},\,1+\tfrac{4}{3}\bigr)=\bigl(5,\tfrac{7}{3}\bigr) \]

5.Method 2: average the three vertices

You do not need a special right-triangle formula once the vertices are known. Gₓ = (x₁ + x₂ + x₃)/3 and Gᵧ = (y₁ + y₂ + y₃)/3. For the example: (3 + 9 + 3)/3 = 15/3 = 5 and (1 + 1 + 5)/3 = 7/3. That is the same (5, 7/3). Translation and averaging must agree; if they do not, a sign or a swapped vertex is wrong.

Type A(3, 1), B(9, 1), C(3, 5) into the Centroid of a Triangle Calculator. G x ≈ 5.00 and G y ≈ 2.33. Use this method whenever the vertices are already listed — it does not care which corner is the right angle, and it does not care whether the legs follow the axes. The shortcut is only a convenience when you have been given legs and a right-angle location instead of three points. Why the two formulas are the same identity is How the Right-Triangle Centroid Relates to the Three-Vertex Formula.

\[ G=\left(\frac{3+9+3}{3},\frac{1+1+5}{3}\right)=\left(5,\tfrac{7}{3}\right) \]

6.When the legs are not parallel to the axes

If the right angle is at A but AB and AC are tilted, you cannot add (b/3, h/3) as axis components. Those components assume the legs are horizontal and vertical. Adding them to A walks the wrong direction.

Example: A(1, 1), B(5, 3), C(0, 3). Vectors AB = (4, 2) and AC = (−1, 2) have dot product zero, so the right angle is at A, but neither leg is axis-aligned. The average is G = ((1+5+0)/3, (1+3+3)/3) = (2, 7/3). A + (b/3, h/3) with b = |AB| and h = |AC| is a different point. In this situation skip the shortcut: use the three vertices, by hand or in the calculator.

\[ G=\frac{A+B+C}{3}\quad\text{(any placement)} \]

7.Plot it, then check the other guides

Once you have G in the given coordinates, plot it the same way as How to Plot the Centroid of a Right Triangle on the Coordinate Plane — axes, the actual vertices, then the translated G, not (b/3, h/3) at the origin. To confirm G on a median, see How to Check a Right-Triangle Centroid with Medians (2:1 Ratio). Legs only, with the right angle still to place: How to Get Centroid Coordinates of a Right Triangle from Legs b and h. The three-vertex tool is always the Centroid of a Triangle Calculator.