Perpendicular Bisector

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Transcript Perpendicular Bisector

EOC Review #5
Thursday
1. Given: Endpoint (2, 3); Midpoint (3, -4)
Find the missing endpoint.
EOC REVIEW QUIZ TOMORROW 
Honors H.W. #23
pg. 246
#2-32 and 38 (even)
H.W. QUESTIONS?
38.) In GHJ, K(2,3) is the midpoint of GH, L(4,1) is
the midpoint of HJ, and M(6,2) is the midpoint of GJ.
Bisectors in Triangles
Toolkit #5.2
Today’s Goal:
1.
To use properties of perpendicular
bisectors and angle bisectors.
Perpendicular Bisector: equidistant from the
endpoints of a segment (cuts SEGMENT in two EQUAL
parts), and makes a RIGHT ANGLE with that segment.

When three perpendicular bisectors meet, that
POINT is called the CIRCUMCENTER.
Find the set of points on the map of D.C.
that are equidistant from the Jefferson
Memorial and the White House.
Ex.1:
CD is the perpendicular bisector of AB.
Find CA and DB.
Angle Bisector: equidistant from the sides of a
triangle – cuts an ANGLE into TWO EQUAL parts.

When three angle bisectors meet, the POINT
is called the INCENTER.
Ex.2(a): Find the value of x,
then find FD and FB.
Ex.2(b):
Check Understanding
a)
According to the diagram, how
far is K from EH? From ED?
b)
What can you conclude about
EK?
c)
Find the value of x.
d)
Find mDEH.
In-Class Practice (Part I)
Geometry Book
Pg. 251
 #’s 1-4, 6-11

Solutions – Part I
1.
AC is the  bis. of BD
2.
15
3.
18
4.
8
6.
x = 12, JK = JM = 17
7.
y = 3, ST = TU = 15
8. HL is the  bis. of
JHG.
9. y = 9, mFHL=54,
mKHL=54
10. 27
11. Point E is on the
bisector of KHF.
In-Class Practice (Part II)
Geometry Book
Pg. 252
 #’s 12-15, 18-25

Solutions – Part II
12. 5
18. 12
13. 10
19. 4
14. 10
20. 4
15. Isosceles
21. 16
22. 5
23. 10
24. 7
25. 14
In-Class Practice (Part III)
Geometry Book
Pg. 252
 #’s 28-30

(Draw picture!)
Solutions – Part III
28. No, A is not equidistant
from the sides of X
29. Yes, AX bisects TXR
30. Yes, A is equidistant from
the side of X.
pg. 253
#40
Bisectors & Graphing
Ch. 5.2 Extension
Today’s Goal(s)
1. To investigate perpendicular and angle
bisectors on the coordinate plane.
Example:
Given points A(1,3), B(5,1), and C(4,4), does C
lie on the  bisector of AB?


Plot points first!
Then determine
whether AC = BC.
#5 Perpendicular Bisector
a. (-2,7) b. (-1,6) c. (0,5)
#6 Angle Bisector
a. (6,5) b. (7,8) c. (4,4)
You are given the points A(4, 8),
O(0,0), and B(12, 0).
What did the waiter say when
he delivered the potato?
What was Humpty-Dumpty’s
Cause of death?
Concurrent Lines,
Medians, and Altitudes
Toolkit 5.3
Today’s Goal:
1.
To identify properties of medians
and altitudes of a triangle.
STOP and THINK!
What does concurrent mean?

When three or more lines intersect in
one point, the point at which they intersect
is the point of concurrency.
Median of a triangle: a segment that
connects a VERTEX to the MIDPOINT of the
opposite side.

When three medians meet, the POINT is called the
CENTROID.
The medians of a triangle are concurrent at a point
where the distance from the point to the vertex is
TWICE the distance from the point to the side.
Ex.1(a): Finding Lengths of Medians
D is the centroid of ABC and DE = 6.
Find BE.
You Try…
Ex.1(b): Finding Lengths of Medians
M is the centroid of WDR, and WM = 16.
Find WX.
Altitude of a triangle: a segment that is
PERPENDICULAR to a side, and connects a
VERTEX to the opposite side.

When three altitudes meet, the POINT is
called the ORTHOCENTER.
4 Points of Concurrency Summary
Helpful Hint!
CM
IN
C
O
PB
AB
M
A
Other Uses!
Perpendicular Bisectors
Pt. of Concurrency:
The circle is circumscribed
about ABC.
Other Uses!
Angle Bisectors
Pt. of Concurrency:
The circle is inscribed
in ABC.
Other Uses!
Medians
Pt. of Concurrency:
“Balance Point”
Also called the center of gravity of a
triangle because it is the point where
a triangular shape will balance.
Other Uses!
Altitudes
Pt. of Concurrency:
“Heights” of a triangle.
Ex.2(a.): Find the center of the
circle that circumscribes XYZ.
You Try…
Ex.2(b.): Find the center of the
circle that circumscribes XYZ.
X(0,0), Y(0,6), Z(4,0)