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
$(-2, -3)$
$(2, 3)$ Cevap
D
$(1, 2)$
E
$(9, -4)$

Cevap

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)

Adım Adım Çözüm

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

Anahtar Kavram

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

Soru

Cevap

Adım Adım Çözüm

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