In the xy-plane, triangle ABC has vertices A(1, 2), B(4, 2), and C(1, 6). The triangle is … — GRE General Test

In the $xy$ -plane, triangle $ABC$ has vertices $A(1, 2)$ , $B(4, 2)$ , and $C(1, 6)$ . The triangle is reflected across the line $y = x$ , and then translated $2$ units to the left and $3$ units downward. What is the $y$ -coordinate of the image of vertex $C$ ?

Answer: -2

Answer

The y-coordinate of the image of vertex C is -2.

Reflecting a point

(x, y)

across the line

y = x

swaps the coordinates, transforming

C(1, 6)

into

(6, 1)

. Subsequently, translating the point

2

units to the left and

3

units downward subtracts

2

from the

x

-coordinate and

3

from the

y

-coordinate, resulting in

(6 - 2, 1 - 3) = (4, -2)

. The

y

-coordinate of this image point is

-2

Step-by-Step Solution

Apply reflection across the line

y = x

to point

C(1, 6)

The transformed point is

C'(6, 1)

Reflecting a point

(x, y)

across the line

y = x

swaps its coordinates to

(y, x)

Apply translation left by

2

units and down by

3

units to

C'(6, 1)

The final point is

C''(6 - 2, 1 - 3) = C''(4, -2)

Translating left subtracts from the

x

-coordinate, and translating downward subtracts from the

y

-coordinate.

Extract the

y

-coordinate of

C''(4, -2)

-2

The

y

-coordinate is the second entry in the coordinate pair

(x, y)

Key Concept

Coordinate Geometry Transformations: Reflection across y = x and Translation

Estimated Time:1m 15s

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