A square in the standard (x, y) coordinate plane has vertices at A(1, 1), B(3, 1), C(3, 3)… — ACT

A square in the standard $(x, y)$ coordinate plane has vertices at $A(1, 1)$ , $B(3, 1)$ , $C(3, 3)$ , and $D(1, 3)$ . The square is dilated by a scale factor of $3$ with the center of dilation at point $A(1, 1)$ . The resulting image is then translated $2$ units left and $4$ units up. What are the coordinates of the final image of vertex $C$ ?

A
$(7, 13)$
B
$(9, 3)$
C
$(1, 19)$
$(5, 11)$ Answer
E
$(11, 13)$

Answer

The final coordinates of the image of vertex

C

are

(5, 11)

To find the coordinates of the image of vertex

C(3, 3)

after the dilation, we calculate the horizontal and vertical distances from the center of dilation

A(1, 1)

C(3, 3)

. Both distances are

2

units. Since the scale factor is

3

, these distances are tripled to

6

units. Adding these to the coordinates of the center

A(1, 1)

gives the dilated point

(1 + 6, 1 + 6) = (7, 7)

. Next, we apply the translation of

2

units left (subtracting

2

from the

x

-coordinate) and

4

units up (adding

4

to the

y

-coordinate) to

(7, 7)

, which yields the final coordinates

(7 - 2, 7 + 4) = (5, 11)

Step-by-Step Solution

Find the coordinates of the image of vertex

C(3, 3)

after a dilation with scale factor

3

centered at

A(1, 1)

The coordinates after dilation are

(7, 7)

The horizontal and vertical distances from the center

A(1, 1)

C(3, 3)

are both

2

units (

3 - 1 = 2

). Multiplying these distances by the scale factor of

3

yields

6

units. Adding these new distances to the coordinates of the center

A(1, 1)

gives

(1 + 6, 1 + 6) = (7, 7)

Apply a translation of

2

units left and

4

units up to the point

(7, 7)

The final coordinates are

(5, 11)

Translating

2

units left subtracts

2

from the

x

-coordinate (

7 - 2 = 5

). Translating

4

units up adds

4

to the

y

-coordinate (

7 + 4 = 11

Key Concept

Transformations in the Coordinate Plane

Estimated Time:1m 30s

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