## Engage NY Eureka Math 8th Grade Module 2 Lesson 5 Answer Key

### Eureka Math Grade 8 Module 2 Lesson 5 Exercise Answer Key

Exercise 1.

Let there be a rotation of d degrees around center O. Let P be a point other than O. Select d so that d≥0. Find P’ (i.e., the rotation of point P) using a transparency.

Answer:

Verify that students have rotated around center O in the counterclockwise direction.

Exercise 2.

Let there be a rotation of d degrees around center O. Let P be a point other than O. Select d so that d<0. Find P’ (i.e., the rotation of point P) using a transparency.

Answer:

Verify that students have rotated around center O in the clockwise direction.

Exercise 3.

Which direction did the point P rotate when d≥0?

Answer:

It rotated counterclockwise, or to the left of the original point.

Exercise 4.

Which direction did the point P rotate when d<0?

Answer:

It rotated clockwise, or to the right of the original point.

Exercises 5–6

Exercise 5.

Let L be a line, \(\overrightarrow{A B}\) be a ray, \(\overline{\boldsymbol{C D}}\) be a segment, and ∠EFG be an angle, as shown. Let there be a rotation of d degrees around point O. Find the images of all figures when d≥0.

Answer:

Verify that students have rotated around center O in the counterclockwise direction.

Exercise 6.

Let \(\overline{A B}\) be a segment of length 4 units and ∠CDE be an angle of size 45°. Let there be a rotation by d degrees, where d<0, about O. Find the images of the given figures. Answer the questions that follow.

Answer:

Verify that students have rotated around center O in the clockwise direction.

a. What is the length of the rotated segment Rotation(AB)?

Answer:

The length of the rotated segment is 4 units.

b. What is the degree of the rotated angle Rotation (∠CDE)?

Answer:

The degree of the rotated angle is 45°.

Exercises 7–8

Exercise 7.

Let L_{1} and L_{2} be parallel lines. Let there be a rotation by d degrees, where -360<d<360, about O.

Is (L_{1} )’∥(L_{2})’?

Answer:

Verify that students have rotated around center O in either direction. Students should respond that (L_{1})’ || (L_{2})’.

Exercise 8.

Let L be a line and O be the center of rotation. Let there be a rotation by d degrees, where d≠180 about O. Are the lines L and L’ parallel?

Answer:

Verify that students have rotated around center O in either direction any degree other than 180. Students should respond that L and L’ are not parallel.

### Eureka Math Grade 8 Module 2 Lesson 5 Exit Ticket Answer Key

Question 1.

Given the figure H, let there be a rotation by d degrees, where d≥0, about O. Let Rotation(H) be H’.

Answer:

Sample rotation shown above. Verify that the figure H’ has been rotated counterclockwise with center O.

Question 2.

Using the drawing above, let Rotation_{1} be the rotation d degrees with d<0, about O. Let Rotation_1 (H) be H”.

Answer:

Sample rotation shown above. Verify that the figure H” has been rotated clockwise with center O.

### Eureka Math Grade 8 Module 2 Lesson 5 Problem Set Answer Key

Question 1.

Let there be a rotation by -90° around the center O.

Answer:

Rotated figures are shown in red.

Question 2.

Explain why a rotation of 90 degrees around any point O never maps a line to a line parallel to itself.

Answer:

A 90-degree rotation around point O will move a given line L to L’. Parallel lines never intersect, so it is obvious that a 90-degree rotation in either direction does not make lines L and L’ parallel. Additionally, we know that there exists just one line parallel to the given line L that goes through a point not on L. If we let P be a point not on L, the line L’ must go through it in order to be parallel to L. L’ does not go through point P; therefore, L and L’ are not parallel lines. Assume we rotate line L first and then place a point P on line L’ to get the desired effect (a line through P). This contradicts our definition of parallel (i.e., parallel lines never intersect); so, again, we know that line L is not parallel to L’.

Question 3.

A segment of length 94 cm has been rotated d degrees around a center O. What is the length of the rotated segment? How do you know?

Answer:

The rotated segment will be 94 cm in length. (Rotation 2) states that rotations preserve lengths of segments, so the length of the rotated segment will remain the same as the original.

Question 4.

An angle of size 124° has been rotated d degrees around a center O. What is the size of the rotated angle? How do you know?

Answer:

The rotated angle will be 124°. (Rotation 3) states that rotations preserve the degrees of angles, so the rotated angle will be the same size as the original.