Transformations on the Coordinate Plane

​Transformations on the Coordinate Plane

​The mirror image produced by flipping a figure over a line is called a reflection.

Because all corresponding sides and angles are congruent, a preimage and its image are said to be congruent after a reflection.

Reflection (Flip)

Multiple Choice

Which axis is triangle ABC reflected over?

• the x-axis

• the y-axis

Multiple Choice

The image of point B is located where?

• (-1, 4)

• (4, -1)

• (1, 4)

• (4, 1)

Triangle ABC is the preimage in this diagram. After the reflection, triangle A'B'C' is called its image.

The image of point B would be point B'. Its coordinates are (-1, 4).

​

​A translation is the movement of a geometric figure in some direction without turning the figure.

Because all corresponding sides and angles are congruent, a preimage and its image are said to be congruent after a translation.

Translation (Slide)

Multiple Select

Select all from below that would describe the translation of triangle ABC to its image A'B'C'.

• 5 units left, 2 units down

• 5 units right, 2 units up

• (x - 5, y - 2)

• (x + 5, y + 2)

Multiple Select

Select all from below that would describe the relationship between the two triangles.

• the angles are congruent

• the side lengths are congruent

• the perimeters are the same

• the areas are the same

• the triangles are congruent

​A rotation occurs when a figure is rotated around a point.

90⁰ = 1/4 turn

180⁰ = 1/2 turn

Because all corresponding sides and angles are congruent, a preimage and its image are said to be congruent after a rotation.

Rotation (Turn)

Multiple Select

Select all from below that would describe the rotation of triangle ABC about the origin.

• 90 degrees clockwise

• 90 degrees counterclockwise

• 180 degrees clockwise

• 180 degrees counterclockwise

The image produced by enlarging or reducing a figure by a scale factor is called a dilation.

Because all corresponding angles are congruent, and side lengths are proportional, a preimage and its image are said to be similar after a dilation.

Dilation

Multiple Select

Select all from below that would describe the relationship between the two triangles.

• the angles are congruent

• the sides are congruent

• the perimeters are congruent

• the areas are congruent

Remember:

​You can use the Pythagorean Theorem to find lengths on the coordinate plane.

Multiple Choice

What kind of transformation was performed from ABCD to A'B'C'D'?

• dilation

• reflection

• rotation

• translation

Multiple Choice

By what scale factor is ABCD dilated?

• $$\frac{1}{4}$$

• $$\frac{1}{2}$$

• 2

• 4

Notice that all the coordinates of A'B'C'D' are twice those of ABCD. This shows that ABCD was dilated by a scale factor of 2.

Multiple Choice

Use the Pythagorean Theorem to find the length of side CD. Round to the nearest tenth.

• 2

• 2.2

• 2.5

• 5

Multiple Choice

Use the Pythagorean Theorem to find the length of side C'D'. Round to the nearest tenth.

• 4.5

• 5.2

• 6

• 10

Multiple Choice

How do the side lengths of CD and C'D' compare? Explain.

• CD is twice as long as C'D' because ABCD was dilated by a scale factor of 2.

• C'D' is twice as long as CD because ABCD was dilated by a scale factor of 2.

• CD is the same length as C'D' because ABCD was dilated by a scale factor of 2.

• There is no relationship between C'D' and CD.

Multiple Choice

A triangle begins in quadrant 2.  It is rotated 90 degrees counterclockwise. 

Which quadrant is it in now?

• 1

• 2

• 3

• 4

Multiple Choice

Select the two transformations that would map ABCD onto A"B"C"D".

• a dilation centered at (0, 0) with a scale factor of $$\frac{1}{2}$$ and then a translation using the rule

  (x + 2, y + 4)

• a dilation centered at (0, 0) with a scale factor of $$\frac{1}{2}$$ and then a reflection over the x-axis

• a dilation centered at (0, 0) with a scale factor of 2 and then a translation of (x - 2, y - 4)

• a dilation centered at (0, 0) with a scale factor of 2 and then a reflection over the x-axis

Multiple Choice

A quadrilateral starts in quadrant 1.  It is reflected over the y-axis and then rotated 180 degrees clockwise.  Which quadrant is it in now?

• 1

• 2

• 3

• 4

Multiple Choice

The perimeter of a triangle is 15 units.  The triangle is translated 8 units left and then reflected over the x-axis. What is the perimeter of the final image of the triangle? 

• There isn't enough information to determine the perimeter.

• The perimeter is 15 units because the triangles are congruent.

• The perimeter is 8 units because it was translated 8 units to the left.

Multiple Choice

Point M(-4, -6) is translated right 5 and down 3.  Then it is rotated 90 degrees counterclockwise.  Which quadrant is it in now?

• 1

• 2

• 3

• 4

Multiple Choice

Select the two transformations that would map ABC onto A"B"C".

• a reflection over the x-axis and then a reflection across the y-axis

• a translation using the rule

  (x + 6, y - 1) and then a reflection over the x-axis

• a reflection over the y-axis and then a 90 degree rotation clockwise about the origin

• a rotation 180 degrees clockwise about the origin and then a translation using the rule

  (x - 1, y + 1)

Step 1: ABC is translated using the rule (x + 6, y - 1) to give image A'B'C'

Step 2: A'B'C' is reflected over the x-axis to give image A"B"C"

Multiple Choice

Point G(4, -3) is reflected over the x-axis and then translated according to the rule (x-8, y-1).  Which quadrant is point G in now?

• 1

• 2

• 3

• 4

Multiple Choice

Select the two transformations that would map ABC onto A"B"C".

• a reflection over the x-axis and then a reflection over the y-axis

• a rotation of 90 degrees clockwise and then a translation using the rule (x, y - 2)

• a reflection over the x-axis and then a rotation of 90 degrees counterclockwise

• a translation using the rule

  (x - 3, y - 5) and then a reflection over the y-axis

Step 1: ABC is reflected over the the x-axis to give image A'B'C'

Step 2: A'B'C' is rotated 90 degrees counterclockwise to give image A"B"C"

Multiple Choice

Point Y(-3, 5) is dilated by a scale factor of 2.  Then it is translated according to the rule (x+3, y-8).  Which quadrant is it in now?

• 1

• 2

• 3

• 4

Multiple Choice

If a figure is rotated 90 degrees clockwise and then reflected over the y-axis, is the final figure congruent to the first?

• YES! Figures that are reflected and/or rotated are always congruent.

• No. After a reflection and/or rotation, the size of the image changes.

Multiple Choice

Select the two transformations that would map ABCD onto A"B"C"D".

• a dilation centered at (0, 0) with a scale factor of 2 and then a reflection over the y-axis

• a dilation centered at (0, 0) with a scale factor of 1.5 and then a reflection over the x-axis

• a dilation centered at (0, 0) with a scale factor of 2 and then a rotation of 180 degrees clockwise around (0, 0)

• a dilation centered at (0, 0) with a scale factor of 1.5 and then a rotation of 180 degrees clockwise around (0, 0)

Step 1: ABCD is dilated centered at (0, 0) with a scale factor of 1.5 to get image A'B'C'D'

Step 2: A'B'C'D' is rotated 180 degrees clockwise about (0, 0) to get A"B"C"D"