Exercise 1 Page 582

In all four options we have to perform a rotation of $90^{\circ }$ counterclockwise about the origin. There are two kinds of options we can choose from. First, by starting with a rotation of $90^{\circ }.$ Second, by starting with a translation.

Starting with a rotation

Let's begin by rotating $△ABC$ by $90^{\circ }$ about the origin. Performing such a transformation means the vertices of the figure change in the following way: Let's perform this rule on the given vertices of $△ABC.$

Point	$(a,b)$	$(\text{-}b,a)$
$A$	$(1,2)$	$(\text{-}2,1)$
$B$	$(3,4)$	$(\text{-}4,3)$
$C$	$(2,2)$	$(\text{-}2,2)$

Now we can draw $A^{\prime }{B}^{\prime }{C}^{\prime }.$

From the diagram, we can see that $A^{\prime }{B}^{\prime }{C}^{\prime }$ has the same orientation as $DEF.$ Additionally, $C^{\prime }$ and $F$ are corresponding vertices. Therefore, if we translate $△A^{\prime }{B}^{\prime }{C}^{\prime }$ $3$ units to the right and $4$ units down, we can map $C^{\prime }$ to $F.$

The transformation we have performed does not match either of the two options where the rotation of $90^{\circ }$ comes first.

Starting with a translation

Let's arbitrarily choose to perform the first transformation, option $B.$ This starts with a translation of 4 units to the left and 3 units down

Using the same rule as previously stated, we can determine the coordinates of the vertices of $A^{\prime \prime }{B}^{\prime \prime }{C}^{\prime \prime }$ after a $90^{\circ }$ rotation about the origin

Point	$(a,b)$	$(\text{-}b,a)$
$A^{\prime }$	$(\text{-}5,\text{-}1)$	$(1,\text{-}5)$
$B^{\prime }$	$(\text{-}1,1)$	$(\text{-}1,\text{-}1)$
$C^{\prime }$	$(\text{-}2,\text{-}1)$	$(1,\text{-}2)$

Now we can draw $A^{\prime \prime }{B}^{\prime \prime }{C}^{\prime \prime }.$

As we can see, option B transforms $△ABC$ to $△DEF.$