Transcript Essential Question: How do I construct inscribed circles

Essential Question:
How do I construct inscribed circles,
circumscribed circles
Standard:
MCC9-12.G.C.3 & 4
Construction #1
Given a triangle construct the circumscribed circle.
Given: Triangle ABC
B
1
3
F
A
8
5
4
PROCEDURE:
7
2
6
1) Construct the perpendicular
bisectors of the sides of the triangle
and label the point of intersection F.
Bisect
From point
segment
B construct
BC; Using
2) Set your compass pointer
ato
arcs
radius
7 F&and
greater
8 the
andradius
draw
thanto a line
point
measure
FC. the
1/2BC
connecting
from
point C
3)
Draw the arcs
circleof
construct
intersections
5with
&
thecenter
6 arcsF ,
C that passes through the vertices
A, B, & C
Now construct
the perpendicular
bisector
of segment
and a3label
Bisectpoint
segment
AC;AB
Using
From
A construct
arcs
&
where
theconnecting
31/2AC
perpendicular
radius
greater
than
from
4point
and F,
draw
a line
the
bisectors
meet.
point C construct
intersections
of thearcs
arcs1 & 2
Making Connections
(Construction #1)
Point F is the circumcenter of the
triangle, because it is the center of the
circle that circumscribes the triangle.
It is equidistant to each of the vertices
of the triangle.
B
F
A
The edges of the triangle ABC are now
chord AB, chord BC, and cord AC of
C circle.
BFC, AFB, and CFA are central
angles of circle F. They are congruent
to the intercepted arcs.
BAC, ACB, and CBA are
inscribed angles of circle F. They are
half the measure of the intercepted arcs.
Construction #2
Given a triangle construct the inscribed circle.
PROCEDURE:
Given: Triangle ABC
1) Construct the angle bisectors of
angles A, B, & C, to get a point of
intersection and call it F.
B
2) Construct a perpendicular to
side AC from point F, and label
this point G.
F
A
X
G
3) Put your pointer on point F and
set your radius to FG.
C
Y
4) Draw the circle using F as the
center and it should be tangent to all
the sides of the triangle.
Making Connections
(Construction #2)
Point F is the incenter of the triangle, because it is the
center of the circle that inscribes the triangle. It is
equidistant to each of the sides of the triangle.
B
The edges of the triangle ABC are now tangent AB,
tangent BC, and tangent AC of circle F.
T
S
F
AS and AU are congrent t angent segments.
BS and BT are congrent t angent segments.
CT and CU are congrent t angent segments.
A
C
U
SFT, TFU, and UFS
are central angles of circle F.
They are congruent to the
intercepted arcs.