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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 = 12
BC =24
Converse 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 Bisectors
Write the equation to solve for x. Justify your equation.
3x + 9 = 7x – 17
Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Find the measure.
mEFH, given that mEFG = 50°.Justify
m EFH = 25
Converse of the Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Write 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!
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Medians 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 2
3 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
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.