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Warm-Up : EOC Prep

Warm-Up : EOC Prep . If m<CTR = 27, what is m<K? A) 27 B) 54 C) 63 D) 76. Suppose RK = 8. What is the perimeter of TPK? 25 33 50 66. Concurrent Lines, Medians, and Altitudes 5.3. Today’s Goals By the end of class today, YOU should be able to….

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Warm-Up : EOC Prep

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  1. Warm-Up:EOC Prep If m<CTR = 27, what is m<K? A) 27 B) 54 C) 63 D) 76 • Suppose RK = 8. What is the perimeter of TPK? • 25 • 33 • 50 • 66

  2. Concurrent Lines, Medians, and Altitudes5.3 Today’s GoalsBy the end of class today, YOU should be able to… 1. Identify and apply the properties of medians and altitudes of a triangle. 2. Find the circumcenter of a triangle.

  3. Concurrency • When three or more lines intersect in one point, they are concurrent. • The point at which they intersect is the point of concurrency. • For any triangle, four different sets of lines are concurrent.

  4. Theorems on concurrency • The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. • The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

  5. Circumcenter • The point of concurrency of the perpendicular bisectors of a triangle

  6. Circumscribed about/Inscribed in • A circle is circumscribed about a triangle if the vertices of the triangle are on the circle. • A circle is inscribed in a triangle if the sides of the triangle are tangent to the circle.

  7. Ex.1: Circumcenters Find the center of the circle that you can circumscribe about OPS.

  8. Ex.1: Solution Two perpendicular bisectors of sides of OPS are x = 2 and y = 3. These lines intersect at (2, 3). This point is the center of the circle.

  9. Incenter • The point of concurrency of the angle bisectors of a triangle • In the following image, points X, Y, and Z are equidistant from I, the incenter.

  10. Median of a triangle • A segment whose endpoints are a vertex and the midpoint of the opposite side.

  11. Theorem 5-8 • 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.

  12. Finding the median of a triangle • D is the centroid of ABC and DE = 6. • Find BE. • Since D is a centroid, BD = BE and DE = BE. 1/3 BE = DE = 6 BE = 18

  13. Centroid of a triangle • The point of concurrency of the medians.

  14. Altitude of a triangle • The perpendicular segment from a vertex to the line containing the opposite side. • Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle or it may lie outside the triangle.

  15. Orthocenter of a triangle • The point of intersection of the lines containing the altitudes of the triangle.

  16. Theorem 5-9 • The lines that contain the altitudes of a triangle are concurrent.

  17. Homework • Page 259 #s 1, 4, 8, 9, 10, 12, 16 • Page 260 #s 19-22, 27 • The assignment can also be found at: • http://www.pearsonsuccessnet.com/snpapp/iText/products/0-13-037878-X/Ch05/05-03/PH_Geom_ch05-03_Ex.pdf

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