Orthocenter & Circumcenter

Name: ______Date: ______Period: ____

Orthocenter & Circumcenter

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 orthocenterThere is an altitude(perpendicular line)

in an acute triangle: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:Right Triangle:Obtuse Triangle:

The orthocenter is locatedThe orthocenter is locatedThe orthocenter is located

______the triangle.______the right angle.______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 thecircumcenter

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:Right Triangle:Obtuse Triangle:

The circumcenteris locatedThecircumcenter is locatedThe circumcenter is located

______the triangle.______the right angle.______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 AP = 20 and BP = , then what is the value of x?

2)The perpendicular bisectors of ΔABC intersect at point P. AP= , BP = 10, and CP = . Find x and y.

3)The perpendicular bisectors of ΔABC are concurrent at P. AP= , BP = and CP = 12. Find x and y.

4)The circumcenter of ΔABC is point P. If AP= , BP = 20, and CP = , find x and y.

5)The circumcenter of ΔJKL is point R. If JR= , KR = , and LR = , 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.

Orthocenter/Circumcenter: Homework

1. The perpendicular bisectors of triangle ABC intersect at point P. If AP= 30 and BP= , then what is the value of x?

2. The perpendicular bisectors of triangle EFG intersect at point P. If EP =22 and FP = , 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.

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?