Question

which rule best describes the transformation used to create △def from △abc?
a. (x, y) → (-x, y)
b. (x, y) → (x + 3, y - 1)
c. (x, y) → (-y, x)
d. (x, y) → (x + 1, y - 3)

Explanation:

Step1: Check option a

The rule $(x,y)\to(-x,y)$ represents a reflection over the y - axis. But the triangles are not related by a y - axis reflection.

Step2: Check option b

Let's take a vertex of $\triangle ABC$, say $A(- 4,-2)$. Applying the rule $(x,y)\to(x + 3,y - 1)$: $x=-4,y = - 2$, then $x+3=-4 + 3=-1$ and $y - 1=-2-1=-3$. But the corresponding vertex $D$ of $\triangle DEF$ is not $(-1,-3)$.

Step3: Check option c

Let's take a vertex of $\triangle ABC$, say $A(-4,-2)$. Applying the rule $(x,y)\to(-y,x)$: $x=-4,y=-2$, then $-y = 2$ and $x=-4$. This is not correct for the transformation.

Step4: Check option d

Take a vertex of $\triangle ABC$, say $A(-4,-2)$. Applying the rule $(x,y)\to(x + 1,y - 3)$: $x=-4,y=-2$, so $x + 1=-4+1=-3$ and $y - 3=-2-3=-5$. For vertex $A(-4,-2)$, the corresponding vertex $D$ of $\triangle DEF$ is $(-3,-5)$. Let's check another vertex, say $B(-1,5)$. Applying the rule $(x,y)\to(x + 1,y - 3)$, we have $x=-1,y = 5$, then $x + 1=-1+1 = 0$ and $y-3=5 - 3=2$. The corresponding vertex $E$ of $\triangle DEF$ is $(0,2)$.

Answer:

D. $(x,y)\to(x + 1,y - 3)$