# Construct a Square and a Nine-Point Circle

### New York State Common Core Math Geometry, Module 1, Lesson 31

Worksheets for Geometry, Module 1, Lesson 31

Student Outcomes

• Students learn to construct a square and begin to construct a nine-point circle.

Construct a Square and a Nine-Point Circle

Classwork

Opening Exercise

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

Exploratory Challenge

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

1. Draw a triangle π΄π΅πΆ.
2. Construct the midpoints of the sides π΄π΅ , π΅πΆ, and πΆπ΄ , and label them as πΏ, π, and π, 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 πΆ to π΄π΅ as π·, the intersection of the altitude from π΄ to π΅πΆΜΜΜΜ as πΈ, and of the altitude from π΅ to πΆπ΄ as πΉ.
5. The altitudes are concurrent at a point; label it π».
6. Construct the midpoints of π΄π», π΅π», πΆπ», and label them π, π, and π, respectively.
7. The nine points, πΏ, π, π, π·, πΈ, πΉ, π, π, π, are the points that define the nine-point circle.

Example

On a blank white sheet of paper, construct a nine-point circle using a different triangle than you used during the notes. Does the type of triangle you start with affect the construction of the nine-point circle?

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