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Daily Warm-Up Quiz

Daily Warm-Up Quiz. Which of your classmates disclosed to a teacher that Mrs. M. sometimes refers to Makenna as Mackenzie…and vice versa? Who told this same teacher that period 2 Geometry is my “favorite class”? How did you determine this?

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Daily Warm-Up Quiz

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  1. Daily Warm-Up Quiz • Which of your classmates disclosed to a teacher that Mrs. M. sometimes refers to Makenna as Mackenzie…and vice versa? • Who told this same teacher that period 2 Geometry is my “favorite class”? How did you determine this? • Who shared that since Monday, Kaylin has been renamed “Kylin”? Geometry

  2. Relationships in Triangles Concurrent Lines, Medians and Altitudes Geometry

  3. Part I: Identifying Properties of AngleBisectors and Perpendicular Bisectors in Triangles Geometry

  4. In this lesson, we will identify properties of perpendicular bisectors and angle bisectors in triangles. ∆ OPS Geometry

  5. Long before the first pencil and paper, some curious person drew a triangle in the sand and bisected the three angles. He noted that the bisectors met in a single point and decided to repeat the experiment on an extremely obtuse triangle. Again, the bisectors concurred. Astonished, the person drew yet a third triangle, and the same thing happened yet again!Unlike squares and circles, triangles have many centers. The ancient Greeks found four: incenter, centroid, circumcenter, and orthocenter. Triangle Centers: http://faculty.evansville.edu/ck6/tcenters/index.html Geometry

  6. Vocabulary & Key Concepts When three or more lines intersect in one point, they are called _____________. The point at which they intersect is called the _________________. concurrent point of concurrency Geometry

  7. Vocabulary and Key Concepts The point of concurrency of the angle bisectors of a triangle is called the _________ of the triangle. I is the incenter of the ∆. incenter THEOREM: The bisectors of the angles of a ∆ are concurrent at a point (incenter) equidistant from the sides. Geometry

  8. City planners want to locate a fountain equidistant from three straight roads that enclose a park. Explain how they can find the location. Checking for Understanding Highway 101 Mariposa Boulevard Andover Road Geometry Check your solution here!

  9. Alert! The common distance is the radius of a circle that passes through the vertices. Vocabulary and Key Concepts The point of concurrency of the perpendicular bisectors of a triangle is called the ____________ of the triangle. circumcenter THEOREM: The perpendicular bisectors of the angles of a ∆ are concurrent at a point (circumcenter) equidistant from the vertices. O is the circumcenter. Geometry

  10. Checking for Understanding: Finding the Circumcenter Checking for Understanding Find the center of the circle that you can circumscribe about ∆ OPS. Solution: Two perpendicular bisectors of the sides of ∆ OPS are x = 2 and y = 3. These lines intersect at (2,3). This point is the center of the circle. Geometry

  11. Homework Geometry

  12. Part II: Identifying Properties of Medians and Altitudes in Triangles Geometry

  13. In this lesson, we will identify properties of medians and altitudes in triangles. ∆ OPS Geometry

  14. Median of a Triangle A median of a triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side. Vertex Midpoint Geometry

  15. Vocabulary and Key Concepts G is the centroid. The point of concurrency of the medians of a triangle is called the___________ of the triangle. centroid FG = 2/3 FC EG = 2/3 EB AG = 2/3 AD Theorem: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. Geometry

  16. Checking for Understanding Finding the Lengths of Medians. G is the centroid of ∆ ABC and DG = 6. Find AG. G is the centroid. AG = 2/3 AD; DG = 1/3 AD 6 = 1/3 AD 18 = AD Geometry

  17. Altitude of a Triangle: An altitude of a triangle is the segmentfrom a vertex to the line containing the opposite side. Unlike angle bisectors and medians, an altitude can lie inside, on, or outside the triangle. perpendicular Acute Triangle: Interior Altitude Right Triangle: Altitude is a side Obtuse Triangle: Exterior Altitude Geometry

  18. Altitude of a Triangle The lines containing the altitudes of a triangle are concurrent at the orthocenter. Theorem: The lines that contain the altitudes of a triangle are concurrent. Geometry http://www.mathopenref.com/triangleorthocenter.html

  19. Identifying Medians and Altitudes A M B Is CM a median, altitude, or neither? Explain. H Is BH a median, altitude, or neither? Explain. C Geometry

  20. Homework Geometry

  21. Solution: City Planning Dilemma The roads form a triangle around the park. By our new theorem, we know that the __________________ of a triangle are concurrent at a point _________ from the sides. The city planners should find the point of concurrency of the _______________ of the triangle formed and locate the fountain there. bisectors of the angles equidistant bisectors of the angles Click hereto return to the lesson! Geometry

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