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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 ALTITUDESGeometry Chapter 5 Relationships within Triangles
Concurrency of Perpendicular Bisector Theorem
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
When three or more lines intersect at one point
Practice Example
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
Solution
Practice Example
4+x=2x-12
X=16
2x=x+3
X=3
Solve for x
Example#1
5x-4=x+6
X=10/4
Solve for x. Solution
Example#2
Find the circumcenter of DEFG with E(4, 4), F(4, 2), and G(8, 2).
(6,3)
Solve for x
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=2
X=16
Median: A segment from a vertex to the midpoint of the opposite side
Median of a triangles
The medians of ABC are AM BL,and CX.
The centroid is point D.
The point of concurrency of the medians is called the centroid.
Centroids
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.Altitude
Altitude
Median
Neither
Example
Find the coordinates of the orthocenter of ∆ABC.
A(6, 10), B(2, 2), C(10, 2)
Real World Connections
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?
Homework Ticket Out