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1.5 Special Points in TrianglesObjectives: Discover points of concurrency in triangles.Draw the inscribed and circumscribed circles of triangles.Warm-Up:
How many times can you subtract the number 5 from 25?Once. After the first calculation you will be subtracting 5 from 20, then 5 from 15, and so on.
If you divide thirty by a half and add ten, what is the answer?
70. When dividing by fractions, you must invert and multiply.
Vocabulary:Inscribed Circle:
A circle in the inside of a triangle that touches each side at one point.
(a circle is inscribed in a polygon if each side of the polygon is tangent to the circle)
Vocabulary:Circumscribed Circle:
A circle that is drawn around the outside of a triangle and contains all three vertices.
(a circle is circumscribed about a polygon if each vertex of the polygon lies on the circle)
Vocabulary:Concurrent:
Literally “running together” of three or more lines intersecting at a single point.
Vocabulary:Incenter:
The center of a inscribed circle; the point where the three angle bisectors intersect.
(It is equidistant from the three sides of the triangle).
Vocabulary:Circumcenter:The center of a circumscribed circle where the three perpendicular bisectors of the sides of a triangle intersect.
(It is equidistant from the three vertices of the triangle).
Example 1:Label the inscribed circle, circumscribed circle, the incenter, the circumcenter, and points of concurrency in the following figures.
inscribed circlecircumscribed circle
incentercircumcenter
points of concurrency
Example 2:Find the perpendicular bisector of each side of XYZ .
Z
Y
X
Example 3:Find the angle bisectors of each angle of LMN .
L
M
N
Example 4:Find the circumscribed circle of JKL.
J
K
L
To find identify the perpendicular bisectors of the triangles sides.
Example 5:Find the inscribed circle of MNO.
M
N
O
To find identify the angle bisectors of the triangle.
Example 6:Find the circumscribed circle of .
A
B C
To find identify the perpendicular bisectors of the triangles sides.
Example 7:Find the inscribed circle of WXY.
W
X
Y
To find identify the angle bisectors of the triangle.