In the standard (x, y) coordinate plane, a triangle is first reflected across the line y =… — ACT

In the standard $(x, y)$ coordinate plane, a triangle is first reflected across the line $y = -x$ and then translated 4 units to the right and 3 units down. If the vertex $A$ of the triangle is mapped to the final image point $A''(1, -5)$ under this composite transformation, what were the coordinates of the original vertex $A$ ?

A
$(8, -5)$
B
$(-2, -3)$
$(2, 3)$ Answer
D
$(1, 2)$
E
$(9, -4)$

Answer

The original coordinates of vertex

A

are

(2, 3)

To find the original coordinates of the vertex, the transformations must be undone in reverse order. First, reverse the translation of 4 units right and 3 units down by moving the final image point

A''(1, -5)

4 units left and 3 units up, yielding the intermediate point

A'(-3, -2)

. Second, reverse the reflection across

y = -x

by applying the rule

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

A'

, resulting in the original coordinates

(2, 3)

Step-by-Step Solution

Identify the inverse of each transformation and their correct reverse order. The composite transformation consists of (1) reflection across the line

y = -x

, followed by (2) translation of 4 units to the right and 3 units down. To backtrack from the final image, we must apply the inverse transformations in reverse order: first (1) reverse the translation, then (2) reverse the reflection.

The reverse process is: first translate 4 units left and 3 units up, then reflect across the line

y = -x

Reversing the composite transformation requires reversing both the individual operations and the order of their application.

Apply the inverse translation to the image point

A''(1, -5)

. Reversing a translation of 4 units right and 3 units down means translating 4 units left and 3 units up, represented by the rule

(x, y) \rightarrow (x - 4, y + 3)

The intermediate point is

A' = (1 - 4, -5 + 3) = (-3, -2)

This step determines the coordinates of the vertex after the reflection but before the translation.

Apply the inverse reflection to

A'(-3, -2)

. Since a reflection is its own inverse, we reflect

A'(-3, -2)

across the line

y = -x

using the rule

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

The original point is

A = (-(-2), -(-3)) = (2, 3)

This step yields the final pre-image coordinates of vertex

A

Key Concept

Backtracking a composite transformation in the coordinate plane by applying inverse transformations in reverse order.

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