Mixed Transformations Geometry

Find the values of x and y that will make the polygon a parallelogram.

What is the sum of the value of x and y?

Answer explanation

Parallelograms have diagonals that bisect eachother.

Find the values of x, y and z.

What is their sum?

Answer explanation

Linear pairs are formed by adjacent angles and are supplementary (add to 180 degrees). Then all the other angles with the variables in the diagram are corresponding angles which are congruent.

Which sequence of transformations of a figure will create an image that is NOT congruent to the original figure? Select all that apply.

• Dilation by a factor of 1 followed by a reflection about line 𝑦 = 2x

• Reflection about the line 𝑦 = 𝑥 followed by a rotation of 90° about point (2, 1)

• Translation by 5 units to the left on the 𝑥 −axis followed by a dilation by a factor of 0.5

• Rotation by 180° followed by a reflection about the 𝑦-axis

Answer explanation

Dilations are not rigid motions and therefore do not preserve congruency!

A rotation about Point 𝐴 maps Point 𝐵 to 𝐵′ and Point 𝐶 to 𝐶 ′ . Which statement must be true?

• 𝑚∠C'AB' = 𝑚∠B'AC

• 𝑚∠C'AC = 𝑚∠B'AB

• If Point 𝐵 is (1, −2) then Point 𝐵′ must be (-1, 2)

• If Point 𝐶 is (1, 4) then Point 𝐶′ must be (−4, 1)

Two right triangles are graphed on the coordinate plane. Triangle 𝐴𝐵𝐶 has vertices 𝐴(0, 0), 𝐵(3, 0), and 𝐶(0, 4). Triangle 𝐷𝐸𝐹 has vertices 𝐷(3, 7), 𝐸(3, −2), and 𝐹(−9, −2). Which of the following transformations cannot be used as part of a proof that 𝐴𝐵𝐶 is similar to 𝐷𝐸𝐹?

• Dilate 𝐴𝐵𝐶 with a center at (0, 0) and a scale factor of 3

• Reflect 𝐴𝐵𝐶 over the line 𝑦 = 0

• Rotate 𝐴𝐵𝐶 90° counterclockwise around the origin

• Translate 𝐴𝐵𝐶 using the transformation (𝑥, 𝑦) → (𝑥 + 3, 𝑦 − 2)

Answer explanation

Dilations can be used to prove similarity!

Graph the pre-image and image to eliminate answer choices.

Before rock climbing, Michelle wants to know how high she will climb. She places a mirror on the ground and walks backward until she can see the top of the cliff in the mirror. She drew a sketch of the situation. What is the height of the cliff?

• 16.5

• 17.5

• 20

• 22

Answer explanation

We have similar triangles here, so set up those proportions, cross multiply and solve!

A ladder leans against the wall and reaches a point 24 feet up the wall. The base of the ladder is 6 feet from the wall. To the nearest degree, what angle does the ladder make with the wall?

Answer explanation

Soh - Cah - Toa

Remember, inverses find angle measures! You needed to use the inverse trig ratios to solve.

Given circle C with radius r and circle D with radius s, what is an appropriate first step when proving circle C is similar to circle D?

• Transform circle C with a dilation that has center C and scale factor s

• Transform circle C with a dilation that has center C and scale factor r

• Transform circle C with the translation that maps point C onto point D

• Transform circle C with a rotation that maps a point on circle C to a point on circle D

Answer explanation

Pay attention to the pre-image and image!

To prove circles are similar:

1st - Translate the circle to match up the centers of the circles

2nd - Dilate the circle

Examine the following figure. ***Draw and label the diagram! If 𝐽𝐿 = 12, 𝐾𝑀 = 2𝑥, and 𝐾𝑀 is a perpendicular bisector of 𝐽𝐿 determine which of the following values are correct. Select all that apply.

• 𝑥 = 4

• 𝐽𝐾 = 10

• 𝐾𝑀 = 8

• 𝑀𝐿 = 6

• Area of ∆𝐽𝐾𝐿 = 48

Write a rule for the translation.

•  (x -1, y  + 2)

•  (x -2, y  + 1)

•  (x +2, y  - 1)

• (x +1, y  -  2)