Question

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 \( \Delta PQR \) is the point \( (\alpha, \beta) \), then \( 15(\alpha - \beta) \) is equal to :

• A. 24
• B. 19
• C. 21
• D. 22

Hint:

The centroid of a triangle is the average of the coordinates of its vertices.

Answer 1

The Correct Option is D

Let \( G \) be the centroid of \( \Delta PQR \) formed by \( (1, 3), (3, 1), (2, 4) \). The centroid of a triangle is the average of the coordinates of its vertices: \[ G = \left( \frac{1+3+2}{3}, \frac{3+1+4}{3} \right) = \left( \frac{6}{3}, \frac{8}{3} \right) = \left( 2, \frac{8}{3} \right). \] Next, we find the image of \( G \) under the transformation \( x + 2y = 2 \). The transformation matrix for this is: \[ \alpha - 2 = \frac{-2}{5}, \quad \beta - \frac{8}{3} = \frac{-32}{15} + 2, \] which leads to \( \alpha = -\frac{2}{5} \) and \( \beta = -\frac{24}{15} \). Thus, \( 15(\alpha - \beta) = 15(-\frac{2}{5} - \frac{24}{15}) = 22 \). Thus, \( 15(\alpha - \beta) = \boxed{22} \).

Answer 2