Problem:
An interior designer is creating a custom coffee table for a client. The top of the table is a glass triangle that needs to balance on a single support. If the coordinates of the vertices of the triangle are at (3, 6), (5, 2), and (7, 10), at what point should the support be placed?
Answer:
The triangle has a balance point at centroid. Find the centroid of the triangular coffee table. Use the centroid formula:
\[\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}\]
The centroid of the triangle is:
Let the points be A(3, 6), B(5, 2), and C(7, 10). The midpoint D of BC is:
Note that AD is a line that connects the vertex A and D, the midpoint of BC. The distance from D(6, 6) to A(3, 6) is 6 – 3 or 3 units. If P is the centroid of the triangle ABC, then AC = 2/3(AD). So, the centroid is 2/3(2) or 2 units to the right of A. The coordinates of the centroid (P) are (3 + 2, 6) or (5, 6).
ANSWER: (5, 6)
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