points of concurrency

```Warm- up Type 2 writing and
Construction
• Write your own definition and draw a picture of
the following:
• Angle Bisector
• Perpendicular Bisector
• Draw an acute triangle. Make the sides at least
three inches in length.
• Construct the perpendicular bisector of one of
the sides of the triangle.
5.3: Concurrent Lines, Medians and
Altitudes, Angle Bisectors,
Perpendicular Bisectors
Objectives:
•To identify properties of perpendicular bisectors and angle
bisectors
•To identify properties of medians and altitudes of triangles
Centroids
• Draw a large acute scalene triangle on your index
card.
• Measure and mark the midpoint of each side of
the triangle.
• Draw a line from each midpoint to the opposite
vertex.
• These three lines should meet at one point!
• Try to balance the triangle by putting your pencil
tip on the point you located.
Median of a triangle – a segment whose endpoints are
the vertex of the triangle and the midpoint of the
opposite side.
Centroid – The point of concurrency of the three medians
of a triangle.
The centroid is always inside the triangle. It is the center
of gravity of the triangle.
Theorem 5-7
The medians of a triangle intersect at a point that is two
thirds of the distance from each vertex to the midpoint of
the opposite side. (The distance from the vertex to the
centroid is twice the distance from the centroid to the
midpoint of the opposite side.)
Median of a Triangle
A segment whose endpoints are a vertex of a
triangle and the midpoint of the opposite side
G
D
BG = 9.21 cm
A
AB = 8.75 cm
DB = 3.44 cm
B
BC = 4.37 cm
BE = 6.88 cm
FB = 4.60 cm
F
E
C
CENTROID: Point of concurrency of medians of a triangle
The medians of a triangle intersect at a point that is 2/3 the
distance from the vertex to the midpoint of the opposite
side.
G is the centroid
AG =
2
3
BG =
2
CG=
2
3
3
BF
CE
O is the centroid of triangle ABC.
CE = 11
FO = 10
CO = 8
1. Find EB.
2. Find OD.
3. Find FB.
4. Find CD.
5. If AO = 6, find OE.
Centroid – The point of concurrency of the medians of
a triangle.
A centroid is always inside the triangle.
A centroid is the center of gravity in a triangle
Concurrent Lines – three or more
lines that intersect in the same
point.
Point of Concurrency – The point of
intersection of three or more lines.
Perpendicular Bisector of a Triangle
– A line ray or segment that is
perpendicular to a side of the
triangle at the midpoint of the side.
A
B
Circumcenter – the point of concurrency of the
perpendicular bisectors of a triangle.
- In an acute triangle the circumcenter is
located inside the triangle.
- In an obtuse triangle the circumcenter is
located outside of the triangle.
-In a right triangle the circumcenter is
on the hypotenuse of the triangle.
Circumcenter:
The point of concurrency of the perpendicular
bisectors of a triangle
 If you were to draw a circle around a triangle,
where each vertex of the triangle are points on the
circle, the circle would be circumscribed about that
triangle
 The circumcenter of the triangle is ALSO the center
of the circle circumscribed about it
Find the center of the circle that you can circumscribe
about the triangle with vertices (0, 0), (-8,0) and (0,6).
1.
2.
3.
4.
Plot the points on a coordinate plane.
Draw the triangle
Draw the perpendicular bisectors of at least 2 sides.
The circumcenter of this triangle will be the center of the
circle.
Perpendicular Bisectors
Acute triangle:
Circumcenter inside
triangle
Right triangle:
Circumcenter lies ON
the triangle
Obtuse Triangle:
Circumcenter is
outside triangle
Perpendicular bisectors of the sides of a triangle
are concurrent at a point equidistant from the
vertices of the triangle
AB = 6.29 cm
AC = 6.29 cm
C
A
B
D
Properties of the Circumcenter
-It is the center of a circle that passes through
the vertices of the triangle
-It is the point that is an equal distance from
each vertex of the triangle.
-In a right triangle it is on the triangle, in an
acute triangle it is inside the triangle, in an
obtuse triangle it is outside of the triangle.
Theorem 5.5 –Concurrency of
Perpendicular Bisectors of a Triangle
The perpendicular bisectors of a
triangle intersect at a point that is
equidistant from the vertices of the
triangle.
Marshfield, Scituate and Hanover want to construct an
indoor pool. They want the pool to be located an equal
distance from the center of each town. Determine where the
pool should be constructed.
Hanover
Scituate
Marshfield
Incenter of a triangle
The point of concurrency of
the three angle bisectors of a
triangle
Theorem 5-6
Concurrency of Angle Bisectors of a Triangle
The angle bisectors of a triangle intersect at a
point that is equidistant from the sides of the
triangle.
The incenter is always located inside the triangle.
m
B
C
A
D
The incenter is the center of a circle that
you can inscribe within the triangle.
When a circle is inscribed inside of a
triangle the circle touches each side of
the triangle at just one point.
B
C
A
D
Altitude of a triangle – a perpendicular
segment from the vertex to the opposite
side or a line that contains the opposite
side.
Orthocenter – the point of concurrency of
the three altitudes of a triangle.
Altitude of a Triangle
The perpendicular segment from the vertex to
the line containing the opposite side
Acute Triangle:
Inside
Right Triangle:
Side
Obtuse Triangle:
Outside
Orthocenter of a Triangle
Point of concurrency for altitudes of a triangle
Acute Triangle:
Inside
Right Triangle:
Vertex
Obtuse Triangle:
Outside
Midsegment of a Triangle – a segment that
connects the midpoints of two sides of a
triangle.
Theorem 5-9 Midsegment Theorem
The segment connecting the midpoints of two
sides of a triangle is parallel to the third side
and is half as long.
Theorem 5-10
If one side of a triangle is longer than another
side, then the angle opposite the longer side is
larger than the angle opposite the shorter side.
Theorem 5-11
If one angle of a triangle is larger than another
angle, then the side opposite the larger angle is
longer than the side opposite the smaller angle.
Theorem 5-12
Exterior Angle Inequality
The measure of an exterior angle of a
triangle is greater than the measure of
either of the two nonadjacent interior
angles.
Theorem 5-13
Triangle Inequality Theorem
The sum of the lengths of any two sides of
a triangle is greater than the length of the
third side.
Hinge Theorem
If two sides of one triangle are congruent to two sides of
another triangle, and the included angle of the first is
larger than the included angle of the second, then the
third side of the first is longer than the third side of the
second.
Converse of the Hinge Theorem
If two sides of one triangle are congruent to two sides of
another triangle, and the third side of the first is longer
than the third side of the second, then the included angle
of the first is larger than the included angle of the second.
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