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Problem 3

If the points $$(2 a, a),(a, 2 a)$$ and $$(a, a)$$ enclose a triangle of area 18 sq. units, find the centroid of the triangle.

Expert verified
The centroid of the triangle is (2,2).
See the step by step solution

Step 1: Formulating the Equation for the Area of the Triangle

The area of the triangle can be calculated using the formula Area = 1/2 |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|. Substituting the given points into the formula, we get 1/2 |2a(2a - a) + a(a - 2a) + a(a - 2a)| = 18.

Step 2: Solving for $$a$$

Solving this equation, we find that $$a$$ equals 3 or -3. However, since $$a$$ cannot be negative in this context, we conclude that $$a$$ equals 3.

Step 3: Calculating the Centroid of the Triangle

The centroid of a triangle is calculated using the formula (x1+x2+x3/3, y1+y2+y3/3). Substituting the values of $$a$$ and the coordinates, we get the centroid as (2,2).

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