Hi! I'm confused on how to solve this math question. Please help me ASAP!

Question

A teacher gave a triangular plate, who vertices on the x-y plane are at (5,8), (2,4), and (8,3), and a stick to a student. The teacher asked the student to balance the plate on the stick. Can you help the student to locate that point on the x-y plane which can balance the plate?

Answer 1

The centroid of the triangle is the center point that will balance the triangular plate. To find the centroid, you must draw at least two of the medians and find where they intersect (their intersection point is the centroid). A median connects a vertex and the midpoint of the opposite side,

Beginning at the vertex (5, 8), the opposite side has endpoints (2, 4) and (8, 3). The midpoint of that opposite side is found using the midpoint equation x_midpt = (x₁ + x₂)/2 = (2 + 8)/2 = 5 and y_midpt = (y₁ + y₂)/2 = (4 + 3)/2 = 3.5 so the midpoint is (5, 3.5). So the median is the line that connects the vertex (5, 8) and the midpoint (5, 3.5). Since these both have the same x-value, the line is x = 5.

Picking another vertex (2, 4), the opposite side has endpoints (5, 8) and (8, 3) making the midpoint (13/2, 11/2) or (6.5, 5.5). The median then is the line having points (2, 4) and (6.5, 5.5). To find that line, first get the slope: m = (y₂ - y₁)/(x₂ - x₁) = ((5.5 - 4)/(6.5 - 2) = 1.5/4.5 = 1/3. Using the point-slope form: y - y₁ = m(x - x₁) with m = 1/3 and (x₁, y₁) = (2, 4) we get the equation of the median line as:

y - 4 = (1/3)(x - 2)

Now, using the info we got from the first median line that x = 5, we can find the y-value of the intersection point:

y - 4 = (1/3)(5 - 2)

y - 4 = (1/3)(3)

y - 4 = 1

y = 5

So the centroid is the point (5, 5)