Download P1 Aim 10 - Manhasset Schools

Aim #10: How do we construct a circumcenter and an incenter?
CC Geometry H
Do Now: Construct the perpendicular bisector of each side.
360°
122
B
A
C
When three or more lines intersect in a single point, they are
concurrent, and the point of intersection is the point of concurrency.
In the diagram above,
• label D, E, and F, the midpoints of AB, BC and AC, respectively.
• mark the right angles
• mark the congruent segments.
• label P, the circumcenter
1. P is equidistant from A and B since it lies on the perpendicular bisector of ____.
2. P is equidistant from B and C since it lies on the perpendicular bisector of ____.
3. P is equidistant from A and C since it lies on the perpendicular bisector of ____.
4. Therefore, ____ = ____ = ____
5. Draw circle P with radius PA.
The intersection of the three perpendicular bisectors of a triangle is the
circumcenter, the center of the circumscribed circle drawn above. The
circumscribed circle is a circle that passes through every vertex of a triangle.
The circumcenter can be located inside, outside or on the triangle.
Using your compass and straightedge, construct the circumcenter Q of ΔABC.
Then draw the circumscribed circle. ∆ABC is a(n) __________ triangle, therefore
the circumcenter is located __________ of the circle.
A
C
B
138°
118
How can you find the center of a circle, by construction, if the center is not shown?
The intersection of the threeangle bisectors of a triangle is theincenter,
the center of the inscribed circle. The inscribed circle is inside the triangle,
and touches each of the sides in exactly one point.
Q of ΔABC.
1. Using your compass and straightedge, construct the incenter
43
B
A
138°
C
2. State your steps for the above construction.
3. Any point on an angle bisector is _____________________ from the sides
forming the angle. Since Q is on the angle bisector of ABC, it is
_________________ from ___ and ___. Similarly, since Q is on the angle bisector
of BCA, it is ___________________ from ____ and ____. Therefore, Q must
also be equidistant from ___ and ___, since it lies on the bisector of BAC. Point Q
is the point of ___________________ of the three angle bisectors.
• The point of concurrency of the angle bisectors must be inside the triangle.
• The point of concurrency of the angle bisectors is equidistant from the three
sides when the perpendicular segment is drawn.
4. Point A is the _____________________ of ΔJKL.
Point B is the ____________________of ΔRST.
5.
Construct the incenter and the inscribed circle.
201
136°
Let's Sum it Up!!
The incenter of a triangle is the center of the circle that is inscribed in that
triangle.
Website to review construction of incenter:
http://www.mathsisfun.com/geometry/construct­triangleinscribe.html
The circumcenter of a triangle is the center of the circle that circumscribes
that triangle.
Website to review construction of circumcenter:
http://www.mathsisfun.com/geometry/construct­trianglecircum.html
Name ______________________
Date ________________
1a) Given line segment AB, use a
compass and straightedge to construct
the set of points that are equidistant
from A and B.
CC Geometry H
HW #10
b) Using a compass and straightedge,
construct the set of points equidistant
from the sides of the angle.
A
B
2a) Construct the circumcenter P of ΔXYZ.
Z
b) Name a fact about point P.
c) Construct circumscribed circle P.
X
Y
3a) Construct the incenter I of ΔQRS.
b) Name a fact about point I.
c) Construct inscribed circle I.
Q
R
S
OVER
Review:
1. For each of the following, construct a line perpendicular to segment AB that
goes through point P.
A
P
A
P
B
B
2. Find x. Then classify the triangle by itsangles and sides.
a)
x = __________
2x
b)
x = __________
960
_____________
_____________
5x
540
_____________
3. Find the value of x that would make a ll b.
x
480
a
b
_____________
4x + 20
5x - 2
2
0
4. The measures of the base angles of an isosceles triangle are (x- 8x + 10) and
2
0
(2x - 5x) . Find the measure of the vertex angle.