Eureka Math Geometry Module 1 Lesson 31 Answer Key

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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.

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

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.

$Eureka Math Geometry Module 1 Lesson 31 Exploratory Challenge Answer Key 2$

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.

$Eureka Math Geometry Module 1 Lesson 31 Exploratory Challenge Answer Key 3$

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

$Eureka Math Geometry Module 1 Lesson 31 Exploratory Challenge Answer Key 4$

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

$Eureka Math Geometry Module 1 Lesson 31 Exploratory Challenge Answer Key 5$

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:

$Eureka Math Geometry Module 1 Lesson 31 Problem Set Answer Key 6$

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:

$Eureka Math Geometry Module 1 Lesson 31 Exit Ticket Answer Key 7$