Name: ________________________________________
Date: ______________
Orthocenter & Circumcenter
Period: ____
1) Concurrency of the Altitudes:
An ___________________of a triangle is a line segment that is drawn from the vertex to the opposite side and is
perpendicular to the side. There are three altitudes in a triangle.
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 can fall in the interior of the triangle, on the side of the triangle, or in the exterior of the triangle.
Acute Triangle:
The orthocenter is located
____________________the triangle.
Right Triangle:
The orthocenter is located
_______ the right angle.
Obtuse Triangle:
The orthocenter is located
_________________________ the triangle.
2) Concurrency of the Perpendicular Bisectors:
The _________________________________________of a triangle is a line segment that is perpendicular (forms a right angle)
and passes through the midpoint of a side of a triangle. There are three perpendicular bisectors in a triangle (one
through each side).
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 point of concurrency of the perpendicular bisectors is known as the
_____________________________ of the triangle.
Construction of the circumcenter
in an acute triangle:
Name: ________________________________________
Date: ______________
Period: ____
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.
Right Triangle:
The circumcenter is located
_______ the right angle.
Obtuse Triangle:
The circumcenter is located
_________________________ the triangle.
Properties of the Circumcenter:
The circumcenter is the center of the circle that can be circumscribed around the triangle.
Example:
Examples:
1) The perpendicular bisectors of ΔABC intersect at point P. If A P = 20 and BP = 2𝑥 + 4, then what is the value
of x?
2) The perpendicular bisectors of ΔABC intersect at point P. AP = 5 + 𝑥, BP = 10, and CP = 2𝑦. Find x and y.
Name: ________________________________________
Date: ______________
Period: ____
3) The perpendicular bisectors of ΔABC are concurrent at P. AP = 2𝑥 − 4, BP = 𝑦 + 6, and CP = 12. Find x and
y.
4) The circumcenter of ΔABC is point P. If AP = 𝑥 + 2𝑦, BP = 20, and CP = 𝑥 + 4, find x and y.
5) The circumcenter of ΔJKL is point R. If JR = 3𝑥 − 5, KR = 2𝑥 + 4, and LR = −2𝑦 + 6, find the value of x and
y.
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: ______________
Orthocenter/Circumcenter: Homework
Period: ____
1. The perpendicular bisectors of triangle ABC intersect at point P. If AP= 30 and BP= 3𝑥 − 6, then what is the
value of x?
𝑥
4
2. The perpendicular bisectors of triangle EFG intersect at point P. If EP =22 and FP = − 19 , 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.
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.
Name: ________________________________________
Date: ______________
Period: ____
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.
Review from yesterday:
How will the fencing alter with the addition of a third home?