## Engage NY Eureka Math 8th Grade Module 3 Mid Module Assessment Answer Key

### Eureka Math Grade 8 Module 3 Mid Module Assessment Task Answer Key

Question 1.

Use the figure below to complete parts (a) and (b).

Answer:

a. Use a compass and ruler to produce an image of the figure with center O and scale factor r=2.

b. Use a ruler to produce an image of the figure with center O and scale factor r=\(\frac{1}{2}\).

Question 2.

Use the diagram below to answer the questions that follow.

Let D be the dilation with center O and scale factor r>0 so that Dilation(P)=P’ and Dilation(Q)=Q’.

a. Use lengths |OQ|=10 units and |OQ’ |=15 units to determine the scale factor r of dilation D. Describe how to determine the coordinates of P’ using the coordinates of P.

Answer:

|OQ|r= |OQ’|

r = \(\frac{15}{10}\)

r = \(\frac{3}{2}\)

The same factor is r = \(\frac{3}{2}\)

Since the coordinates of P = (-4, -3) the coordinates of the dilated point P’ will be the same factor times the coordinates of P. Therefore P’ = (\(\frac{3}{2}\) × (-4), \(\frac{3}{2}\) × (-3)) = (-6, -45)

b. If |OQ|=10 units, |OQ’|=15 units, and |P’Q’|=11.2 units, determine |PQ|. Round your answer to the tenths place, if necessary.

Answer:

\(\frac{15}{10}\) = \(\frac{11.2}{|PQ|}\)

15(|PQ|) = 112

|PQ| = \(\frac{112}{15}\) ≈ 7.5

The length of \(\overline{P Q}\) is about 7.5 units.

Question 3.

Use a ruler and compass, as needed, to answer parts (a) and (b).

a. Is there a dilation D with center O that would map figure PQRS to figure P’Q’R’S’? Explain in terms of scale factor, center, and coordinates of corresponding points.

Answer:

P = (3, -2)

P’ = (9, -6)

Q = (3, -1)

Q’ = (9, -3)

R = (4, 1)

R’ = (12, 3)

S = (0, 2)

S’ = (0, 6)

Yes, there is a dilation D with Genter O, that maps PQS to P’Q’R’S’. The scale factor is 3. The image of each point is 3 times the coordinates of the original image. For example, P = (3, -2) and P’ = (3 × 3, 3 × (-2)) = (9, -6)

b. Is there a dilation D with center O that would map figure PQRS to figure P’Q’R’S’? Explain in terms of scale factor, center, and coordinates of corresponding points.

Answer:

No, there is not a dilation D that will map PQRS to P’Q’R’S’. A dilation will move A point, S, to its image S’ on the ray \(\overrightarrow{O S}\). In the picture above O, S, S’ are not on the same ray. A similar statement can be made for points P, Q and R, Therefore, there is no dilation that maps PQRS to P’Q’R’S’.

c. Triangle ABC is located at points A(-4,3), B(3,3), and C(2,-1) and has been dilated from the origin by a scale factor of 3. Draw and label the vertices of triangle ABC. Determine the coordinates of the dilated triangle A’B’C’, and draw and label it on the coordinate plane.

Answer:

A= (-4, 3)

A’ = (-4×3, 3×3) = (-12, 9)

B =(3, 3)

B’ = (3×3, 3×3) = (9, 9)

C = (2, -1)

C’ = (2×3, -1×3) = (6, -3)