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 triangle PQR is the point (alpha, beta), then 15(alpha – beta) is equal to : (1) 24 (2) 19 (3) 21 (4) 22

Key Concepts Needed

1. Reflection of a Point in a Line
   A line in standard form is $Ax + By + C = 0$.
   For any point $(x_0, y_0)$ its mirror image $(x', y')$ is

   $$d = \frac{Ax_0 + By_0 + C}{A^2 + B^2}$$ $$x' = x_0 - 2A d, \qquad y' = y_0 - 2B d$$

   The idea is: measure the signed perpendicular distance $d$ from the point to the mirror, then walk the same distance on the other side (hence the "minus twice" part).

2. Centroid of a Triangle
   If a triangle has vertices $(x_1, y_1),\,(x_2, y_2),\,(x_3, y_3)$, its centroid is just the average of coordinates:

   $$\bigl(\,\frac{x_1 + x_2 + x_3}{3},\; \frac{y_1 + y_2 + y_3}{3}\,\bigr)$$

   Because each corner weighs the same, the arithmetic mean is the balance point.

3. Why each step?
   We reflect first because the new vertices decide the new centroid.
   We average second because the centroid formula is simplest after we know all three image points.
   Finally, substituting into $15(\alpha - \beta)$ just follows the statement of the problem.

Imagine you have a tiny triangle drawn on a sheet of paper.
You put a mirror (the line $x + 2y = 2$) on the paper. Every point of the triangle shines into the mirror and makes an image point on the other side—just like your face in a mirror.

Now, instead of worrying about every little detail, we care about the balance point (the centroid) of the new reflected triangle. Think of the centroid as where you would place a pin so the triangle could spin like a top without falling over.

The question finally asks: “After you find that balance point, do a simple subtraction ($\alpha - \beta$) and multiply by 15. What number pops out?”

So the game plan is:

1. Reflect each original corner across the mirror-line.
2. Average the $x$ values and the $y$ values—that gives the centroid.
3. Plug those two numbers ($\alpha$ and $\beta$) into $15(\alpha - \beta)$.

That final number is one of the four options.

Below is a complete, step-by-step derivation.

1. Write the mirror line in standard form

2. Reflect each vertex

Let $(x_0, y_0)$ be a vertex; compute

then

Vertex $A(1, 3)$

So $P(-1, -1)$.

Vertex $B(3, 1)$

So $Q\bigl(\tfrac{9}{5},\, -\tfrac{7}{5}\bigr)$.

Vertex $C(2, 4)$

So $R\bigl(-\tfrac{6}{5},\, -\tfrac{12}{5}\bigr)$.

3. Find the centroid $(\alpha, \beta)$ of $\triangle PQR$

4. Compute $15(\alpha - \beta)$

5. Match with the options

The correct value is 22, i.e. option (4).