Construct a Square and a Nine-Point Circle

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

Worksheets for Geometry

Student Outcomes

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

Construct a Square and a Nine-Point Circle


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.

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 𝐴𝐡𝐢.
  2. Construct the midpoints of the sides 𝐴𝐡 , 𝐡𝐢, and 𝐢𝐴 , 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 𝐢 to 𝐴𝐡 as 𝐷, the intersection of the altitude from 𝐴 to 𝐡𝐢̅̅̅̅ as 𝐸, and of the altitude from 𝐡 to 𝐢𝐴 as 𝐹.
  5. The altitudes are concurrent at a point; label it 𝐻.
  6. Construct the midpoints of 𝐴𝐻, 𝐡𝐻, 𝐢𝐻, and label them 𝑋, π‘Œ, and 𝑍, respectively.
  7. The nine points, 𝐿, 𝑀, 𝑁, 𝐷, 𝐸, 𝐹, 𝑋, π‘Œ, 𝑍, are the points that define the nine-point circle.


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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