In the standard (x, y) coordinate plane, a triangle has vertex A(3, -4). If the triangle i… — ACT

In the standard $(x, y)$ coordinate plane, a triangle has vertex $A(3, -4)$ . If the triangle is reflected across the line $y = x$ and then translated $5$ units to the left and $2$ units up, what are the coordinates of the image of vertex $A$ ?

A
$(1, 1)$
B
$(-2, -2)$
C
$(-2, 6)$
$(-9, 5)$ Answer
E
$(-1, -1)$

Answer

(-9, 5)

To find the coordinates of the image, we apply the transformations in the specified order. First, reflecting the point

(3, -4)

across the line

y = x

swaps the

x

- and

y

-coordinates, resulting in

(-4, 3)

. Second, translating the point

5

units to the left subtracts

5

from the

x

-coordinate, and translating

2

units up adds

2

to the

y

-coordinate:

(-4 - 5, 3 + 2) = (-9, 5)

Step-by-Step Solution

Reflect the point across the line

y = x

(-4, 3)

The coordinate rule for a reflection across the line

y = x

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

. Applying this to the vertex

A(3, -4)

swaps the coordinates to yield

(-4, 3)

Translate the point

5

units to the left and

2

units up

(-9, 5)

Translating a point

5

units left subtracts

5

from the

x

-coordinate, and translating

2

units up adds

2

to the

y

-coordinate:

(-4 - 5, 3 + 2) = (-9, 5)

Key Concept

Applying composite transformations in the coordinate plane including reflections and translations.

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