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5.2: Circumcenters and Incenters

This educational activity focuses on exploring the properties of circumcenters and incenters in various types of triangles. Using patty paper, students will draw isosceles, acute, right, and obtuse triangles, applying concepts such as perpendicular bisectors and angle bisectors. They will learn how to find the circumcenter and incenter of triangles through hands-on practice, including graphing, measuring, and analyzing distances. This activity enhances understanding of geometric properties and will help students recognize the significance of these centers in triangle construction and circle inscribing.

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5.2: Circumcenters and Incenters

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  1. 5.2: Circumcenters and Incenters Objectives: To know and apply the properties of circumcenters. To know and apply the properties of incenters.

  2. Vocabulary:

  3. 5.2: Circumcenters and Incenters Activity: Need: 2 pieces of patty paper. A pencil or felt tip pen (2 colors preferably) A strait-edge (ruler) Glue stick to share with a partner On your patty paper:  Draw an isosceles triangle on each paper  Draw an acute triangle on each paper  Draw a right triangle on each paper  Draw an obtuse triangle on each paper

  4. 5.2: Circumcenters and Incenters Activity: Label each paper:  bisector  bisector

  5. Perpendicular Bisectors of a triangle… • bisect each side at a right angle • meet at a point called the circumcenter • The circumcenter is equidistant from the 3 vertices of the triangle. • The circumcenter is the center of the circle that is circumscribed about the triangle. • The circumcenter could be located inside, outside, or ON the triangle. C

  6. Using the Circumcenter…. Example 1 Find all measures that are possible in the figure.

  7. Example 2: Finding the Circumcenter of a Right Triangle Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6). Step 1 Graph the triangle. Step 2 Draw in two perpendicular bisectors. Step 3 Find the intersection of the 2 lines. Answer: the circumcenter is at (5, 3)! Now complete: page 311 #12 – 17, 20 (10 minutes!)

  8. Now complete: page 311 #12 – 17, 20

  9. Paste-able! Angle Bisectors of a triangle… • bisect each angle • meet at the incenter • The incenter is equidistant from the 3 sides of the triangle. • The incenter is the center of the circle that is inscribed in the triangle. • The incenter is always inside the circle. I

  10. QX and RX are angle bisectors of ΔPQR. Find the distance from X to PQ. Example 1 Find mPQX.

  11. 2.JP, KP, and HP are angle bisectors of ∆HJK. Find the distance from P to HK. Example 2

  12. Now complete: page 311 #12 – 32

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