properties of triangles
DESCRIPTION
Chapter 5. Properties of Triangles. Vocab Review. Intersect Midpoint Angle Bisector Perpendicular Bisector Construction of a Perpendicular through a point on a line. Perpendicular Bisector Theorem. If a point is on the perpendicular bisector of a segment, - PowerPoint PPT PresentationTRANSCRIPT
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PROPERTIES OF TRIANGLES
Chapter 5
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Vocab Review Intersect Midpoint Angle Bisector Perpendicular Bisector
Construction of a Perpendicular through a point on a line
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Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment,
then it is
equidistant from the endpoints of the segment
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Converse of the Perpendicular Bisector
Theorem If a point is equidistant from the endpoints of the segment ,
then it is on the
perpendicular bisector of a segment
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Using Perpendicular Bisectors
What segment lengths are equal?
Why is L on OQ?
Given: OQ is the bisector of MP
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Angle Bisector Theorem
If a point is on the bisector of an angle,
then it is
equidistant from the two side of the angle
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Converse of the Angle Bisector Theorem
If a point is in the interior of an angle AND is equidistant from the sides of the angle,
then it lies on the
bisector of the angle
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Angle Bisector Theorem Let’s Prove
it…
Given: • D is on the
bisector of BAC• BD AB• CD AC
Prove: DB DC
STATEMENTS
REASONS
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“Guided Practice”Page 267 #1-7
CHECK POINT
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Perpendicular Bisector of a Triangle
Line (or ray/ segments) that is perpendicular to a side of the triangle at the midpoint of the side.
Concurrent @ Circumcenter
(not necessarily inside the triangle)
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Perpendicular Bisector of a Triangle
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Concurrency
Point of Concurrency The point of
intersection of the lines
Concurrent Lines When 3 or more
lines (rays/segments) intersect at the same point. .
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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 AG = BG =
CG
Circumcenter
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Exit Ticket
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Angle Bisector of a Triangle
Bisector of an angle of the triangle.(one for each angle)
Concurrent @ Incenter (always inside
the triangle)
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Angle Bisector of a Triangle
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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 EG = DG =
FG
Incenter
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Pythagorean Theorem
a2 + b2 = c2
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Pythagorean Theorem
a2 + b2 = c2
EXAMPLE:a= ?, b= 15, c=
17a2 + b2 = c2
a2 + 152 = 172
a2 + 225 = 289a2 = 64a = 8
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ClassworkPage 275-278
#10-23,
CHECK POINT
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Median of a Triangle
Segment whose end points are a vertex of the triangle and the midpoint of the opposite side.
Concurrent @ Centroid(always inside
the triangle)
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Median of a Triangle
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Concurrency of Medians of a Triangle
The medians of a triangle intersect at a point
that is ²/₃ of the distance from each vertex to the midpoint of the opposite side.AG = ²/₃AD, BG = ²/₃BE, CG
= ²/₃CF
Centoid
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Page 280 Example #1
EXAMPLE
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Altitude of a Triangle
Perpendicular segment from a vertex to the opposite side.
Concurrent @ Orthocenter(inside, on, or outside the
triangle)
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Page 281 Example #3
EXAMPLE
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Concurrency of Altitudes of a Triangle
Lines containing the altitudes of a triangle are concurrent
Intersect at some point (H) called the orthocenter
(Inside, On, or Outside)
orthocenter
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Activity
Video
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Classwork:pg 282 #13-16
Homework:pg 282 #1-11, 17-20, &
34
CHECK POINT
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Warm-upFind the coordinates of the midpoint:
1. (2, 0) and (0, 2)2. (5, 8) and (-3, -4)
Find the slope of the line through the given points:
1. (3, 11) and (3, 4)2. (-2, -6) and (4, -9)3. (4, 3) and (8, 3)
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Midsegment of a Triangle
*Segment that connects the midpoints of two sides of a triangle
*Half as long as the side of the triangle it is parallel to
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** REVIEW **
MIDPOINT FORMULA
Midpoint =
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Midsegment of a Triangle
Find a Midsegment of the triangle….
Let D be the midpoint of AB
Let E be the midpoint of BC
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Midsegment Theorem
The segment connecting the midpoint of two sides of a triangle is parallel to the third side and is half as long
DEBC and DE=¹₂BC⁄
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Prove it…
How do you prove DEBC?
How do you prove DE=¹₂BC?⁄
Use the slope formula to find each slope (parallel lines have the same slope)
Use the distance formula to find the distance of each segment
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Exit TicketPg 290 #3-9
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Classwork:pg 290 #10-18, 21-25
CHECK POINT
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Warm-upIn ∆DEF, G, H, and J are the midpoints of the sides…
J
H
G
F
ED1. Which segment is parallel to EF?
2. If GJ=3, what is HE?3. If DF=5, what is HJ?
GH
3
2.5
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Pgs 310-311#1-15
Pgs 313#1-11
Test Review