Day 4 agenda Go over homework- 5 min Warm-up- 10 min 5.3 notes- 55 min Start homework- 20 min The students will practice what they learned in the computer lab during the following class period.

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? A)25 B)33 C)50 D)66

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

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.

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

Circumcenter The point of concurrency of the perpendicular bisectors of a triangle

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.

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

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.

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.

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

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.

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

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

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.

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

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

Homework Page 259 #s 1, 4, 8, 9, 10, 12, 16 Page 260 #s 19-22, 27 The assignment can also be found at: ts/ X/Ch05/05-03/PH_Geom_ch05- 03_Ex.pdfhttp:// ts/ X/Ch05/05-03/PH_Geom_ch05- 03_Ex.pdf