A vertex of a polygon in the standard (x, y) coordinate plane undergoes two transformation… — ACT

A vertex of a polygon in the standard $(x, y)$ coordinate plane undergoes two transformations: it is first reflected across the line $y = x$ , and then it is translated $4$ units to the right. If the coordinates of the final image of the vertex are $(1, 5)$ , what were the coordinates of the vertex before the transformations?

A
$(1, 1)$
B
$(5, 5)$
$(5, -3)$ Cevap
D
$(9, 1)$
E
$(-3, 5)$

Cevap

The coordinates of the vertex before the transformations were

(5, -3)

To find the pre-image of the vertex, we apply the inverse transformations in reverse order. First, we undo the translation of

4

units to the right by translating the image point

(1, 5)

4

units to the left, resulting in

(-3, 5)

. Second, we undo the reflection across the line

y = x

by swapping the coordinates of

(-3, 5)

, which yields the original point

(5, -3)

Adım Adım Çözüm

Identify the inverse of the second transformation (translation of

4

units to the right).

The inverse is a translation of

4

units to the left, which subtracts

4

from the

x

-coordinate:

(1 - 4, 5) = (-3, 5)

To find the pre-image, we must undo the transformations in reverse order, starting with the last transformation applied.

Identify the inverse of the first transformation (reflection across the line

y = x

The inverse of a reflection across

y = x

is itself, which swaps the

x

- and

y

-coordinates:

(-3, 5) \rightarrow (5, -3)

Undoing the first transformation on the intermediate coordinates gives the original pre-image coordinates.

Anahtar Kavram

Working backward from a final image using inverse transformations in reverse order.

Alternatif Yöntem

We can write the composite transformation as an algebraic rule. A reflection across

y = x

maps

(x, y) \to (y, x)

. A translation

4

units to the right maps

(y, x) \to (y + 4, x)

. Setting the final coordinates equal to

(1, 5)

, we get

y + 4 = 1

(which means

y = -3

) and

x = 5

. Thus, the original coordinates were

(5, -3)

Tahmini Süre:1m 15s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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Alternatif Yöntem

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