Name: ______________________________________________________________
Date: ________________________
Period: ______
Chapter 3: Constructions
Topic 5: Orthocenter, Circumcenter & Circumcircle
Construction #9: Orthocenter - All three ALTITUDES
The orthocenter is found at the intersection of all three ____________________________ in a triangle. To construct
the altitudes, construct a perpendicular line through the side of the triangle from the opposite vertex (use
construction #8b – point off the line).
The altitudes of a triangle, extended if necessary, are concurrent in a point called the ___________________________
of the triangle.
Construction of an orthocenter
in an acute triangle:
There is an altitude(perpendicular line)
drawn from each vertex of the triangle!
Location of the Orthocenter:
The orthocenter is not always located inside the triangle. The location of the circumcenter depends on the
type of triangle that we have.
Acute Triangle:
The orthocenter is located
____________________the triangle.
triangle.
Example:
Right Triangle:
The orthocenter is located
_______ the right angle.
Obtuse Triangle:
The orthocenter is located
_________________________ the
Name: ______________________________________________________________
Date: ________________________
Period: ______
Construction #10: Circumcenter - All three PERPENDICULAR BISECTORS
The circumcenter is found at the intersection of all three _______________________________ ________________________of
a triangle. To construct the circumcenter, construct the perpendicular bisectors (construction #7) to al three
sides.
The perpendicular bisectors of the three sides of a triangle are concurrent in a point that is equidistant (the
same distance) from the vertices of the triangle. The circumcenter of the triangle becomes the center of a
circle known as the
______________________________ which goes around the outside of the triangle.
Construction of the circumcenter
in an acute triangle:
Location of the Circumcenter:
The circumcenter is not always located inside the triangle. The location of the circumcenter depends on the
type of triangle that we have.
Acute Triangle:
The circumcenter is located
_________________ the triangle.
Example:
Right Triangle:
The circumcenter is located
___________________ the right
angle.
Obtuse Triangle:
The circumcenter is located
_________________ the triangle.
Name: ______________________________________________________________
Examples:
Date: ________________________
1) The perpendicular bisectors of ΔABC intersect at point P. If A P = 20 and BP =
value of x?
2) The perpendicular bisectors of ΔABC intersect at point P. AP =
y.
3) The perpendicular bisectors of ΔABC are concurrent at P. AP =
x and y.
, then what is the
, BP = 10, and CP =
, BP =
Period: ______
. Find x and
and CP = 12. Find
Name: ______________________________________________________________
Date: ________________________
4) The circumcenter of ΔABC is point P. If AP =
, BP = 20, and CP =
5) The circumcenter of ΔJKL is point R. If JR =
x and y.
, KR =
, and LR =
Period: ______
, find x and y.
, find the value of
6) The perpendicular bisectors of ΔLMN intersect at point J. LJ = 3x - y, MJ = x + y, and NJ = 4. Find the
value of x and y.
Name: ______________________________________________________________
Date: ________________________
Period: ______
Chapter 3: Constructions
Topic 5 Homework: Orthocenter, Circumcenter & Circumcircle
1. The perpendicular bisectors of triangle ABC intersect at point P. If AP= 30 and BP=
the value of x?
2. The perpendicular bisectors of triangle EFG intersect at point P. If EP =22 and FP =
, then what is
, then what is
the value of x?
3. The circumcenter of triangle ABC is point P. If AP = 4x - 6, BP = 3x + 3, and CP = -2y + 6, find the value of x
and y.
Name: ______________________________________________________________
Date: ________________________
Period: ______
4. The circumcenter of triangle ABC is point P. If AP = 3m + 15, BP = -5m - 9, and CP = -2y + 12, find the value
of x and y.
5. Two homes are built on a plot of land. Both homeowners have dogs, and are interested in putting up as
much fencing as possible between their homes on the land, but in a way that keeps the fence equidistant
from each home. Use your construction tools to determine where the fence should go on the plot of land.
(Think what the shape will look like… a straight line equidistant from both homes… start by connecting the
two houses with a light line and think from there…)
How will the fencing alter with the addition of a third home?