Holt Geometry 5-1 Perpendicular and Angle Bisectors
Feb 23, 2016
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Find each measure of MN. Justify
MN = 2.6
Perpendicular Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Write an equation to solve for a. Justify
3a + 20 = 2a + 26
Converse of Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Find the measures of BD and BC. Justify
BD = 12BC =24Converse of Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Find the measure of BC. Justify
BC = 7.2
Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle BisectorsWrite the equation to solve for x. Justify your equation.
3x + 9 = 7x – 17
Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle BisectorsFind the measure.
mEFH, given that mEFG = 50°.Justify
m EFH = 25
Converse of the Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle BisectorsWrite an equation in point-slope form for the perpendicular bisector of the segment with endpoints C(6, –5) and D(10, 1).
Perpendicular Bisectors of a triangle…
• bisect each side at a right angle• meet at a point called the circumcenter• The circumcenter is equidistant from the 3 vertices of
the triangle. • The circumcenter is the center of the circle that is
circumscribed about the triangle. • The circumcenter could be located inside, outside, or
ON the triangle.
C
Angle Bisectors of a triangle…
• bisect each angle• meet at the incenter• The incenter is equidistant from the 3
sides of the triangle.• The incenter is the center of the circle that
is inscribed in the triangle. • The incenter is always inside the circle.
I
Paste-able!
DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC.
GC = 13.4
GM = 14.5
MZ and LZ are perpendicular bisectors of ∆GHJ. Find GM
GK = 18.6 JZ = 19.9
Z is the circumcenter of ∆GHJ. GK and JZ
Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6).
MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN.
MP and LP are angle bisectors of ∆LMN. Find mPMN.
mPMN = 30
Holt Geometry
5-1 Perpendicular and Angle BisectorsMedians of triangles:• Endpoints are a vertex and midpoint of opposite side.• Intersect at a point called the centroid • Its coordinates are the average of the 3 vertices.
• The centroid is ⅔ of the distance from each vertex to the midpoint of the opposite side.
• The centroid is always located inside the triangle.
5-3: Medians and Altitudes
P
A Z
YX
C
B
2 2 23 3 3
AP AY BP BZ CP CX
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Altitudes of a triangle:• A perpendicular segment from a vertex to
the line containing the opposite side.• Intersect at a point called the
orthocenter. • An altitude can be inside, outside, or on
the triangle.
5-3: Medians and Altitudes
In ∆LMN, RL = 21 and SQ =4. Find LS.
LS = 14
In ∆LMN, RL = 21 and SQ =4. Find NQ.
12 = NQ
In ∆JKL, ZW = 7, and LX = 8.1. Find KW.
KW = 21
Example 2: Problem-Solving Application
A sculptor is shaping a triangular piece of iron that will balance on the point of a cone. At what coordinates will the triangular region balance?
Find the average of the x-coordinates and the average of the y-coordinates of the vertices of ∆PQR. Make a conjecture about the centroid of a triangle.
Find the orthocenter of ∆XYZ with vertices X(3, –2), Y(3, 6), and Z(7, 1).
X