A triangle with vertices (2,3), (4,3) and (3,5) undergoes the transformation (x, y) → (−x, −y). Which transformation is this: a translation, a reflection across the y-axis, a 180 degree rotation about the origin, or a reflection across the x-axis?

Step-by-step solution

1. 1

   Compare the rule with the standard rules. Each basic transformation has its own coordinate signature:

   $\text{reflect in }y\text{-axis}:(x,y)\to(-x,y);\quad \text{reflect in }x\text{-axis}:(x,y)\to(x,-y)$

   $\text{rotate }180^{\circ}\text{ about }O:(x,y)\to(-x,-y);\quad \text{translate}:(x,y)\to(x+h,y+k)$

   The given rule negates both coordinates, which matches only the third.

2. 2

   Rule out the translation. A translation adds the same fixed vector to every point. Here $(2,3)\to(-2,-3)$ shifts by $(-4,-6)$ while $(3,5)\to(-3,-5)$ shifts by $(-6,-10)$ — different vectors, so it is not a translation.

3. 3

   Rule out both reflections. A reflection across the $y$-axis would leave $y$ unchanged, sending $(2,3)$ to $(-2,3)$; a reflection across the $x$-axis would leave $x$ unchanged, sending it to $(2,-3)$. Neither equals the actual image $(-2,-3)$.

4. 4

   Confirm the half-turn geometrically. For any point $P=(x,y)$, its image $P'=(-x,-y)$ satisfies: the origin is the midpoint of $PP'$, since $\tfrac{x+(-x)}{2}=0$ and $\tfrac{y+(-y)}{2}=0$; and $|OP'|=\sqrt{x^{2}+y^{2}}=|OP|$. Same distance, opposite direction — that is precisely a rotation of $180^{\circ}$ about the origin.

   $\boxed{\text{C: a }180^{\circ}\text{ rotation about the origin, e.g. }(2,3)\to(-2,-3)}$

5. 5

   Apply it to the whole triangle and note a property. The image vertices are $(-2,-3)$, $(-4,-3)$ and $(-3,-5)$. A half-turn is a rigid motion that preserves lengths and angles but, unlike a reflection, preserves orientation — the vertices still run in the same rotational order. It is also its own inverse: applying it twice returns every point to its start.