Bellringer Use the figure at the right for Exercises 1–4. 1. What is the relationship between LN and MO? Perpendicular bisector 2. What is the value of x? X=10 3. Find LM. 4. Find LO. 3. LM= 50 4.LO=50

Bellwork- MA.912.G.1.3 In the figure below, AB is parallel to DC. Which of the following statements about the figure must be true?   A. m∠DAB + m∠ABC = 180° B. m∠DAB + m ∠ CDA = 180° C. ∠BAD is congruent to ∠ADC D. ∠ADC is congruent to ∠ABC Highlands Park is located between two parallel streets, Walker Street and James Avenue. The park faces Walker Street and is bordered by two brick walls that intersect James Avenue at point C, as shown below What is the measure, in degrees, of ∠ACB, the angle formed by the park’s two brick walls?

5-3 Bisectors in Triangles and 5.4 Medians and Altitudes Geometry Chapter 5 Relationships within Triangles

Review A point on the perpendicular bisector is equidistant to the endpoints of the segment. A point on the angle bisector is also equidistant to the angle sides.

Lesson Purpose Objective Essential Question To identify properties of perpendicular and angle bisectors. How is the intersection of angle bisectors in a triangle different from the intersection of perpendicular bisectors? The angle bisectors in a triangle meet at a single point that is equidistant from the sides of the triangle. This point is the incenter of an inscribed circle instead of circumscribed about the triangle .

Concurrency of Perpendicular Bisector Theorem When three or more lines intersect at one point The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter of the triangle, which is equidistant from the vertices of the triangle. Point of Concurrency of a Perpendicular bisectors of a triangle

Practice Example RE=AE 3(x+3)=21 3x+9=21 3x=21-9→3x=12 X=4 AN=KN Solution RE=AE 3(x+3)=21 3x+9=21 3x=21-9→3x=12 X=4 AN=KN 4y-3=9→4y=9+3 4y=12 Y=3

Practice Example Solve for x 4+x=2x-12 X=16 2x=x+3 X=3

Example#1 Solve for x. Solution 5x-4=x+6 X=10/4

Example#2 Solve for x Find the circumcenter of DEFG with E(4, 4), F(4, 2), and G(8, 2). (6,3) Need to do the distance formula for each line, √(0-0)²+(0-8)²= 8 √(0-10)²+(0-8)²=2√41 √(0-10)²+(8-8)²=10

Concurrency of Angle Bisectors Theorem The angle bisectors of a triangle intersect at a point called the incenter of the triangle, which is equidistant from the sides of the triangle. Point of concurrency of the angle bisectors of a triangle

Practice Example Find the value of x? The angle bisectors intersect at P. The incenter P is equidistant from the sides, so SP = PT. Therefore, x = 9. Note that , the continuation of the angle bisector, is not the correct segment to use for the shortest distance from P to .

Examples Find the value of x? X=14 X=16 X=2

Median of a triangles Median: A segment from a vertex to the midpoint of the opposite side

Centroids The medians of ABC are AM BL,and CX. The point of concurrency of the medians is called the centroid. The centroid is point D.

Altitude of a triangle Altitude: The perpendicular segment from a vertex to the line that contains the opposite side. In an acute triangle all of the altitudes are all inside the triangle.

Altitudes of Right and Obtuse Triangle In a right triangle, two of the altitudes are the legs of the triangle and the third is inside the triangle. In an obtuse triangle, two of the altitudes are outside of the triangle.

Orthocenter The orthocenter is the point of concurrency for the altitudes. The altitudes of ∆QRS are AM, BL, and CX. The orthocenter is point V.

Examples Determine whether is a median, an altitude, or neither.

Example Find the coordinates of the orthocenter of ∆ABC.

Real World Connections

Summary Meet at a single point Perpendicular Bisectors in Triangles Angle Bisector in Triangles Meet at a single point That is equidistant from the vertices of triangle Point of concurrency is circumcenter of the triangle Meet at a single point that is equidistant from the sides of triangle Point is incenter of an inscribed circle Circle is inscribed in triangle

Ticket Out and Homework Sect. 5-3 pg. 319 #'s 4,7,11 Pg. 322 #'s 4-8,10 Sect. 5-4 Pg327-8 #’s 7- 10,14,15 Pg331-2#’s 7-9,12 What is true of the segments that connect the incenter and circumcenter to each side of the triangle?