Let the triangle PQR be the image of the triangle with vertices (1,3),(3,1) and (2,4) in the line x+2y=2. If the centroid of △ PQR is the point (α,β), then 15(α-β) is equal to :

Solution

Maths sir · 4 steps

Do not reflect three points. Reflect just the centroid, because a reflection carries it along.

1. Centroid first. A reflection maps the centroid of a triangle to the centroid of its image. The original centroid is ((1+3+2)/3, (3+1+4)/3) = (2, 8/3).

   (α - x₁)/a = (β - y₁)/b = -2(ax₁ + by₁ + c)/(a² + b²)

2. Put in numbers. For x + 2y - 2 = 0, a = 1, b = 2, c = -2. Then ax₁ + by₁ + c = 2 + 16/3 - 2 = 16/3, so the common ratio is (-2 × 16/3)/5 = -32/15.
3. Find the image. α = 2 - 32/15 = -2/15 and β = 8/3 - 2 × 32/15 = (40 - 64)/15 = -24/15.
4. So. α - β = (-2 + 24)/15 = 22/15, so 15(α - β) = 22. The answer is (c).

The idea to keep. Reflections, rotations and translations keep the centroid as the centroid. To find the image triangle's centroid, find the original centroid and transform just that one point.

Answer from the NTA final answer key. Solution by PrepWiser.