Download Mod 1 - Aim #10 - Manhasset Public Schools

Aim #10: How do we construct a circumcenter and an incenter?
CC Geometry H
Do Now: Construct the perpendicular bisector of each side.
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 triangle.
A
C
B
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.
1. Using your compass and straightedge, construct the incenter Q of ΔABC.
B
A
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.
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.