A triangle has vertices K(-2, 3), L(-2, 1), and M(2, 1) in the standard (x, y) coordinate … — ACT

A triangle has vertices $K(-2, 3)$ , $L(-2, 1)$ , and $M(2, 1)$ in the standard $(x, y)$ coordinate plane. The triangle is first reflected across the line $y = x$ , and then the reflected image is translated $3$ units to the right and $4$ units down. What are the coordinates of the final image of vertex $K$ ?

A
$(0, 2)$
B
$(1, -1)$
$(6, -6)$ Answer
D
$(0, -2)$
E
$(5, -1)$

Answer

(6, -6)

The correct answer is

(6, -6)

. To find this, start with the coordinates of vertex

K(-2, 3)

. First, apply the reflection across the line

y = x

, which maps

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

. This gives the intermediate coordinates

K'(3, -2)

. Second, apply the translation

3

units to the right and

4

units down, which maps

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

. Adding

3

to the

x

-coordinate and subtracting

4

from the

y

-coordinate of the intermediate point yields the final coordinates

(3 + 3, -2 - 4) = (6, -6)

Step-by-Step Solution

Apply the reflection rule to the pre-image vertex

K(-2, 3)

K'(3, -2)

Reflecting a point

(x, y)

across the line

y = x

swaps its coordinates, mapping it to

(y, x)

Apply the translation rule to the reflected point

K'(3, -2)

K''(6, -6)

Translating a point

3

units to the right and

4

units down adds

3

to the

x

-coordinate and subtracts

4

from the

y

-coordinate.

Key Concept

Composite transformations in the coordinate plane

Question

Answer

Step-by-Step Solution

Key Concept

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