Lesson 31: Construct a Square and a Nine-Point Circle
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 31
M1
GEOMETRY
Lesson 31: Construct a Square and a Nine-Point Circle Classwork 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.
Lesson 31: Date:
Construct a Square and a Nine-Point Circle 10/15/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.168 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 31
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
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
2.
Construct the midpoints of the sides ̅̅̅̅̅, ̅̅̅̅ , and ̅̅̅̅̅, and label them as ,
3.
Construct the perpendicular from each vertex to the opposite side of the triangle (each is called an altitude). Label the intersection of the altitude from to ̅̅̅̅ as , the intersection of the altitude from to ̅̅̅̅ as , and of the altitude from to ̅̅̅̅ as .
4. 5.
. , and , respectively.
6.
The altitudes are concurrent at a point, label it . Construct the midpoints of ̅̅̅̅ , ̅̅̅̅ , ̅̅̅̅ and label them , , and , respectively.
7.
The nine points, ,
, , , , , , , , are the points that define the nine-point circle.
Lesson 31: Date:
Construct a Square and a Nine-Point Circle 10/15/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.169 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 31
M1
GEOMETRY
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?
Lesson 31: Date:
Construct a Square and a Nine-Point Circle 10/15/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.170 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 31
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Problem Set Construct square a.
and square
so that
b.
Each side of is half the length of each ̅̅̅̅ contains ̅̅̅̅ .
c.
The midpoint of ̅̅̅̅ is also the midpoint of ̅̅̅̅ .
Lesson 31: Date:
.
Construct a Square and a Nine-Point Circle 10/15/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.171 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 31
M1
GEOMETRY
Lesson 31: Construct a Square and a Nine-Point Circle Classwork 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.
Lesson 31: Date:
Construct a Square and a Nine-Point Circle 10/15/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.168 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 31
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
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
2.
Construct the midpoints of the sides ̅̅̅̅̅, ̅̅̅̅ , and ̅̅̅̅̅, and label them as ,
3.
Construct the perpendicular from each vertex to the opposite side of the triangle (each is called an altitude). Label the intersection of the altitude from to ̅̅̅̅ as , the intersection of the altitude from to ̅̅̅̅ as , and of the altitude from to ̅̅̅̅ as .
4. 5.
. , and , respectively.
6.
The altitudes are concurrent at a point, label it . Construct the midpoints of ̅̅̅̅ , ̅̅̅̅ , ̅̅̅̅ and label them , , and , respectively.
7.
The nine points, ,
, , , , , , , , are the points that define the nine-point circle.
Lesson 31: Date:
Construct a Square and a Nine-Point Circle 10/15/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.169 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 31
M1
GEOMETRY
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?
Lesson 31: Date:
Construct a Square and a Nine-Point Circle 10/15/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.170 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 31
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Problem Set Construct square a.
and square
so that
b.
Each side of is half the length of each ̅̅̅̅ contains ̅̅̅̅ .
c.
The midpoint of ̅̅̅̅ is also the midpoint of ̅̅̅̅ .
Lesson 31: Date:
.
Construct a Square and a Nine-Point Circle 10/15/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.171 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
