Hello vjesus12.

Let's take a look at your question.

You are correct, you have to use the formulas $$2X+2Y=46\ \ and\ \ \ \ \ X\cdot Y=76 .$$ Hence, $$X+Y=23\ \ and\ \ \ \ \ X=\frac{76}{Y}.$$ Replacing X on the first equation we get $$\frac{76}{Y}+Y=23\ \Leftrightarrow\ \ \ 76+Y^2=23Y\ \Leftrightarrow\ \ Y^2-23Y+76=0$$ $$Y^2-23Y+76=0\ \Leftrightarrow\ \ \left(Y-4\right)\left(Y-19\right)=0.$$ Hence, Y=4 or Y=19.

If Y=4 then X+Y=23 implies that X=23-4=19. In this case, the dimensions of the rectangle are 4 and 19.

If Y=19 then X+Y=23 implies that X=23-19=4. In this case, the dimensions of the rectangle are 4 and 19.

In each case, the length of the shorter side is equal to 4.

This is why the correct answer is the option A.

I hope this answer can help you.

Feel free to ask me again if you have a doubt.

Regards.