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Find the center of the circle that circumscribes XYZ .

Warm Up 1) What is the difference between a perpendicular bisector and an angle bisector? 2) True or False: A segment bisects another segment and creates four right triangles.

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Find the center of the circle that circumscribes XYZ .

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  1. Warm Up1) What is the difference between a perpendicular bisector and an angle bisector?2) True or False: A segment bisects another segment and creates four right triangles.

  2. Paper Folding Bisectorspage 272 baby blue sectionshare a book with your partner1) Make a conjecture about the bisectors of the angles of a triangle.2) Make a conjecture about the perpendicular bisectors of the sides of a triangle.

  3. Concurrent- When three or more lines intersect in one point Point of Concurrency- The point at which they intersect3) Is there any relationship between the lengths of the point of concurrency and the vertex or sides of the triangle?

  4. Theorems:The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. (Gizmo Application )

  5. Find the center of the circle that circumscribes XYZ. Concurrent Lines, Medians, and Altitudes LESSON 5-3

  6. The lines y = 4 and x = 3 intersect at the point (3, 4). This point is the center of the circle that circumscribes XYZ. Concurrent Lines, Medians, and Altitudes LESSON 5-3 Additional Examples (continued) Vocab: Circumcenter of a triangle Circumscribed about a triangle Quick Check

  7. Circumcenter of a Triangle- The point of concurrency of the perpendicular bisectors of a triangleCircle is circumscribed about the triangle. Incenter of a Triangle- The point of concurrency of the angle bisectors of a triangleWe can inscribe a circle in the triangle.

  8. Median of a TriangleIs a segment whose endpoints are a vertex and the midpoint of the opposite side. Are the medians of one triangle concurrent? If so, is there a relationship between the median and a point of concurrency? Centroid- Point of Concurrency of the medians

  9. M is the centroid of WOR, and WM = 16. Find WX. 2 3 Because M is the centroid of WOR, WM = WX. 2 3 WM = WXTheorem 5-8 2 3 16 = WXSubstitute 16 for WM. 3 2 24 = WXMultiply each side by . Concurrent Lines, Medians, and Altitudes LESSON 5-3 Additional Examples The centroid is the point of concurrency of the medians of a triangle. The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. (Theorem 5-8) Quick Check

  10. Altitude of a Triangleis the perpendicular segment from a vertex to the line containing the opposite side. Theorem: The lines that contain the altitudes of a triangle are concurrent. In your notes, draw the altitude of a triangle for an Acute Triangle, Right Triangle and Obtuse Triangle.

  11. Is KX a median, an altitude, neither, or both? Because LX = XM, point X is the midpoint of LM, and KX is a median of KLM. Because KX is perpendicular to LM at point X, KX is an altitude. So KX is both a median and an altitude. Concurrent Lines, Medians, and Altitudes LESSON 5-3 Additional Examples Quick Check

  12. Practice 5-3#1, 2, 5-11

  13. Assignment • P275-277 • #1-4, 9, 12-16, 19-22

  14. Find the relationship Create a completed triangle using straws. Try different length straws Find a relationship between the lengths of the sides that complete a triangle and that would not complete a triangle. What conclusion can you draw about the smallest angles and largest angles with respect to the sides?

  15. What do we notice….. Triangle inequality Theorem. The sum of any two side lengths of a triangle is greater than the third side length. A AB + BC > AC B C BC + AC > AB AC + AB > BC

  16. Examples Tell whether a triangle can have sides with the given lengths. Explain. • 3, 5, 7 2. 4, 6, 11

  17. Workbook • Page 347

  18. Comparison Property of InequalityIf a = b + c and c > 0 then a > b Corollary to the Triangle Exterior Angle TheoremThe measure of an exterior angle of a triangle is greater than the measure of each of its remote interior anglesm <1 > m<2 and m<1 > m<3If two sides of a triangle are not congruent, then the larger angle lies opposite the longer sideIf two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

  19. Assignment Due Monday, 11/28 • Textbook Page 292 #1-27 odd • Try to get this done in class, ideally your homework is to enjoy this break with family and friends!

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