POINTS OF CONCURRENCYPOINTS OF CONCURRENCY

In this lesson we will define what a point of concurrency is.Then we will look at 4 points of concurrency in triangles.

As you go through the powerpoint, you will complete yournotesheet.

You will need to be able to define the 4 points of concurrency and identify them in a picture. You will alsouse the definition to identify relationships between Segments and angles to solve problems.

When two lines intersect at one point, we say that the lines are intersecting. The point at which they intersect isthe point of intersection.

Well, if three or more lines intersect, we say that the linesare concurrent. The point at which these lines intersectIs called the point of concurrency.

(nothing new right?)

POINTS OF CONCURRENCYPOINTS OF CONCURRENCY

The perpendicular bisectors of the sides of a triangle areconcurrent at a point equidistant from the vertices.

Point of concurrency

Perpendicular bisectors=CircumcenterPerpendicular bisectors=Circumcenter

This point of concurrency has a special name. It is known as the circumcenter of a triangle.

Circumcenter

Perpendicular bisectors=CircumcenterPerpendicular bisectors=Circumcenter

The circumcenter is equidistant from all three vertices.

Perpendicular bisectors=CircumcenterPerpendicular bisectors=Circumcenter

The circumcenter gets its name because it is the center of the circle that circumscribes the triangle. Circumscribe means to be drawn around by touching as many points as possible.

Perpendicular bisectors=CircumcenterPerpendicular bisectors=Circumcenter

Sometimes the circumcenter will be inside the triangle, sometimes it will be on the triangle, and sometimes it will be outside of the triangle!

Acute Right Obtuse

Perpendicular bisectors=CircumcenterPerpendicular bisectors=Circumcenter

The bisectors of the angles of a triangle areconcurrent at a point equidistant from the sides.

Angle bisector

Angle bisectors=IncenterAngle bisectors=Incenter

The point of concurrency of the three angle bisectors is another center of a triangle known as the Incenter. It is equidistant from the sides of the triangle, and gets its name from the fact that it is the center of the circle that is inscribed within the circle.

Angle bisectors=IncenterAngle bisectors=Incenter

Incenter

Median: A median of a triangle is the segment That connects a vertex to the midpoint of the opposite side.

Medians=CentroidMedians=Centroid

This point of concurrency of the medians is another center of a triangle. It is known as the Centroid.

Centroid

Medians=CentroidMedians=Centroid

This Centroid is also the Center of Gravity of a triangle which means it is the point where a triangular shape willbalance.

Medians=CentroidMedians=Centroid

Altitudes of a triangle are the perpendicularsegments from the vertices to the line containingthe opposite side.

Unlike medians, and angle bisectors that arealways inside a triangle, altitudes can be inside, on or outside the triangle.

Altitudes=OrthocenterAltitudes=Orthocenter

This point of concurrency of the altitudes of a triangle form another center of triangles. This center is known as the Orthocenter.

Altitudes=OrthocenterAltitudes=Orthocenter

The Orthocenter of a triangle can be insideon or outside of the triangle.

There are many Points of Concurrency in Triangle. Wehave only looked at four:

Circumcenter: Where the perpendicular bisectors meet

Incenter: Where the angle bisectors meet

Centroid: Where the medians meet

Orthocenter: Where the altitudes meet.

POINTS OF CONCURRENCYPOINTS OF CONCURRENCY

Practice ProblemsPractice ProblemsOn the half sheet of paper that you were givencomplete the following problems.

1A-C Identify the point of concurrency that is shown in the triangle

A CB

Practice ProblemsPractice Problems

Solve for x, y in each triangle using the given information

2. Point G is a centroidAC = 24, AF=15, AE= 3x-6, BF = 3y

3. Point H is an orthocenter<ABE = 25, <BFC = 8x + 10, <BAE= 6y+5

Practice ProblemsPractice Problems

4. Point W is both a centroid and an orthocenter.Why is it a circumcenter also?

5. If you draw two medians, is that enough to determine where the centroid of a triangle is? Why/Why not?

6. Do the following matching.A.Circumcenter W. Angle bisectorsB.Incenter X. MediansC.Orthocenter Y. Perpendicular bisectorsD.Centroid Z. Altitudes