In the standard (x, y) coordinate plane, a point is first dilated by a scale factor of 2 w… — ACT

In the standard $(x, y)$ coordinate plane, a point is first dilated by a scale factor of $2$ with the center of dilation at $(1, 1)$ . The resulting intermediate point is then reflected across the line $y = -x$ to yield a final image at $(-5, -9)$ . What were the coordinates of the original point before these two transformations were applied?

A
$(17, 9)$
B
$(4, 2)$
$(5, 3)$ Answer
D
$(-4, -2)$
E
$(-2, -4)$

Answer

The original point before the transformations was

(5, 3)

To find the coordinates of the original point, we reverse the transformations in reverse order. First, we reverse the reflection across

y = -x

. Since reflecting

(x', y')

across

y = -x

results in

(-y', -x') = (-5, -9)

, we have

y' = 5

and

x' = 9

, which makes the intermediate point

(9, 5)

. Next, we reverse the dilation centered at

(1, 1)

with a scale factor of

2

. Under this dilation, the original point

(x, y)

maps to

(1 + 2(x - 1), 1 + 2(y - 1)) = (9, 5)

. Solving the equation

1 + 2(x - 1) = 9

gives

x = 5

, and solving

1 + 2(y - 1) = 5

gives

y = 3

. Thus, the original point is

(5, 3)

Step-by-Step Solution

Determine the intermediate point by reversing the reflection across the line

y = -x

The final image is at

(-5, -9)

. The rule for reflection across

y = -x

(x', y') \rightarrow (-y', -x')

. To reverse this, we apply the same rule to the final image:

x' = -(-9) = 9

and

y' = -(-5) = 5

. Thus, the intermediate point is

(9, 5)

We must work backward from the final result, reversing the second transformation first.

Set up the equation to reverse the dilation centered at

(1, 1)

with a scale factor of

2

The dilation formula for a point

(x, y)

centered at

(h, k)

with scale factor

c

(x', y') = (h + c(x - h), k + c(y - k))

. Substituting

h=1

k=1

c=2

x'=9

, and

y'=5

yields the equations:

9 = 1 + 2(x - 1)

and

5 = 1 + 2(y - 1)

The intermediate point is the result of dilating the original point, so we solve for the original coordinates.

Solve the coordinate equations to find the original values of

x

and

y

For

x

8 = 2(x - 1) \Rightarrow 4 = x - 1 \Rightarrow x = 5

. For

y

4 = 2(y - 1) \Rightarrow 2 = y - 1 \Rightarrow y = 3

. The original point is

(5, 3)

Solving these algebraic equations gives the horizontal and vertical coordinates of the pre-image.

Key Concept

Transformations in the Coordinate Plane

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