In the standard (x, y) coordinate plane, a triangle undergoes a sequence of transformation… — ACT

In the standard $(x, y)$ coordinate plane, a triangle undergoes a sequence of transformations. First, it is reflected across the line $y = -x$ . Then, the resulting figure is translated $3$ units to the right and $2$ units down. If the vertex $A''$ of the final image is located at $(-1, 5)$ , what were the coordinates of the original vertex $A$ ?

$(-7, 4)$ Answer
B
$(-3, -2)$
C
$(7, -4)$
D
$(4, -7)$
E
$(-4, 7)$

Answer

The original coordinates of vertex

A

are

(-7, 4)

The coordinates of the original vertex are found by reversing the sequence of transformations. Starting from the final image at

(-1, 5)

, we first reverse the translation by moving

3

units to the left and

2

units up, which gives the intermediate point

(-4, 7)

. Then, we apply the reflection across the line

y = -x

to this intermediate point. The reflection rule swaps the coordinates and negates them, mapping

(-4, 7)

(-7, 4)

Step-by-Step Solution

Identify the forward transformation rules for a point

(x, y)

The reflection across

y = -x

maps

(x, y) \rightarrow (-y, -x)

. The translation

3

units right and

2

units down maps

(x', y') \rightarrow (x' + 3, y' - 2)

Understanding the forward transformations allows us to set up the equations or work backward systematically.

Work backward from the final image

A''(-1, 5)

by reversing the translation.

Reversing a translation of

3

units right and

2

units down means translating

3

units left and

2

units up. Applying this to

A''(-1, 5)

yields

(-1 - 3, 5 + 2) = (-4, 7)

This gives the intermediate coordinates of the point

A'

before the translation occurred.

Reverse the reflection across the line

y = -x

A reflection across

y = -x

is its own inverse. Applying the mapping

(x', y') \rightarrow (-y', -x')

to the intermediate point

(-4, 7)

yields

(-7, 4)

This step recovers the original pre-image coordinates of vertex

A

Key Concept

Reversing composite transformations in the coordinate plane

Estimated Time:1m 30s

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