chapter 5. vocab review intersect midpoint angle bisector perpendicular bisector construction...
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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,
then it is
equidistant from the endpoints of the segment
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
Using Perpendicular Bisectors
What segment lengths are equal?
Why is L on OQ?
Given: OQ is the bisector of MP
Angle Bisector Theorem
If a point is on the bisector of an angle,
then it is
equidistant from the two side of the angle
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
Angle Bisector Theorem Let’s Prove
it…
Given: • D is on the
bisector of BAC• BD AB• CD AC
Prove: DB DC
STATEMENTS
REASONS
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)
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. .
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
Angle Bisector of a Triangle
Bisector of an angle of the triangle.(one for each angle)
Concurrent @ Incenter (always inside
the triangle)
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
Pythagorean Theorem
a2 + b2 = c2
EXAMPLE:a= ?, b= 15, c=
17a2 + b2 = c2
a2 + 152 = 172
a2 + 225 = 289a2 = 64a = 8
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)
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
Altitude of a Triangle
Perpendicular segment from a vertex to the opposite side.
Concurrent @ Orthocenter(inside, on, or outside the
triangle)
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
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)
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
Midsegment of a Triangle
Find a Midsegment of the triangle….
Let D be the midpoint of AB
Let E be the midpoint of BC
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⁄
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
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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