## Engage NY Eureka Math Geometry Module 1 Lesson 31 Answer Key

### Eureka Math Geometry Module 1 Lesson 31 Opening Exercise Answer Key

Opening Exercise:

With a partner, use your construction tools and what you learned in Lessons 1 – 5 to attempt the construction of a square. Once you are satisfied with your construction, write the instructions to perform the construction.

Answer:

Steps to construct a square:

- Extend line segment \(\overline{A B}\) on either side of A and B.
- Construct the perpendicular to \(\overline{A B}\) through A; construct the perpendicular to \(\overline{A B}\) through B.
- Construct circle A: center A, radius AB; construct circle B: center B, radius BA.
- Select one of the points where circle A meets the perpendicular through A, and call that point D. In the same half plane as D, select the point where B meets the perpendicular through B, and call that point C.
- Draw segment CD.

### Eureka Math Geometry Module 1 Lesson 31 Exploratory Challenge Answer Key

Exploratory Challenge:

Now, we are going to construct a nine-point circle. What is meant by the phrase nine-point circle?

Answer:

A circle that contains a set of nine points.

Steps to construct a nine-point circle:

1. Draw a triangle ABC.

2. Construct the midpoints of the sides \(\overline{A B}\), \(\overline{B C}\), and \(\overline{C A}\), and label them as L, M, and N, respectively.

3. Construct the perpendicular from each vertex to the opposite side of the triangle (each is called an altitude).

4. Label the intersection of the altitude from C to \(\overline{A B}\) as D, the intersection of the altitude from A to \(\overline{B C}\) as E, and of the altitude from B to \(\overline{C A}\) as F.

5. The altitudes are concurrent at a point; label it H.

6. Construct the midpoints of \(\overline{A H}\), \(\overline{B H}\), \(\overline{c H}\). and label them X, Y, and Z, respectively.

7. The nine points, L, M, N, D, E, F, X, Y, Z, are the points that define the nine-point circle.

### Eureka Math Geometry Module 1 Lesson 31 Example Answer Key

Example 1:

On a blank white sheet of paper, construct a nine-point circle using an equilateral or right triangle then you used during the notes. What do you notice?

Answer:

Some of the key points in the nine-point circle construction of a right triangle or equilateral triangle will coincide with each other.

### Eureka Math Geometry Module 1 Lesson 31 Problem Set Answer Key

Question 1.

Construct square ABCD and square GHIJ so that

a. Each side of GHIJ is half the length of each ABCD.

b. \(\overline{A B}\) contains \(\overline{G H}\).

c. The midpoint of \(\overline{A B}\) is also the midpoint of \(\overline{G H}\).

Answer:

Sample construction might look like the following:

### Eureka Math Geometry Module 1 Lesson 31 Exit Ticket Answer Key

Question 1.

Construct a square ABCD and a square AXYZ so that \(\overline{A B}\) contains X and \(\overline{A D}\) contains Z.

Answer:

Sample construction might look like the following: