Objectives: To identify properties of perpendicular bisectors and angle bisectors

Points of Concurrency

Points of concurrency

• Concurrent lines are three or more lines that intersect at the same point. The mutual point of intersection is called the point of concurrency.

Perpendicular Bisectors

Angle Bisectors

Medians Altitudes

Perpendicular Bisectors

Angle Bisectors

Medians Altitudes

Peanut Butter CookiesAre Best InMilk Chocolate And Ovaltine

Perpendicular Bisectors - CircumcenterAngle Bisector - IncenterMedians - Centroid Altitude - Orthocenter

Perpendicular Bisectors

circumcenter

vertex

Construct the perpendicular bisectors of each side of the

triangle. **NOT always inside the triangle. **

Perpendicular Bisectors

circumcenter

vertex

Construct the perpendicular bisectors of each side of the

triangle.

Angle Bisectors

incenter

side

**ALWAYS inside the triangle. **

Construct the angle bisectors from each vertex of the triangle.

The incenter is the center of the triangle's incircle, the largest circle that will fit inside

the triangle and touch all three sides

Angle Bisectors

incenter

side

Construct the angle bisectors from each vertex of the triangle.

Example #1

The perpendicular bisectors of ∆ABC meet at point D.  a. Find DB

 b.Find AE

  c. Find ED (Hint: Use the Pythagorean Theorem.) Write your answer in simplified radical form.

Example #2

R is the circumcenter of ∆OPQ. OS = 10, QR = 12, and PQ = 22. a.Find OP b. Find RP c. Find OR d. Find TP e. Find RT

Example #3

 Your family is considering moving to a new home. The diagram shows the locations of where your parents work and where you go to school. The locations form a triangle.  In this diagram, how could you find a point that is equidistant from each location? Explain your answer.        

homework

• 1. Centers of Triangles Worksheet

**Be prepared for a 10 minute Pop Quiz on today’s lesson next time. [10 points] ** It will be on circumcenter and incenter of a triangle.

Medians

medianvertex

midpoint

“center of gravity”Construct the median of each

side of the triangle.

**Always INSIDE the triangle. **

The centroid is exactly two-thirds the way along each median.

Put another way, the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.

Medians

medianvertex

midpoint

“center of gravity”Construct the median of each

side of the triangle.

Steps to find the centroid:1. Find the midpoint of each side of the triangle.2. Draw a segment joining the vertex of the triangle to the midpoint of the oppositeside.3. The point of intersection of the three segments is the centroid.

Construct the three altitudes of the triangle

**NOT always inside the triangle. **

Obtuse Acute Right

Altitudes

altitude

perpendicular

Altitudes

altitude

perpendicular

Construct the three altitudes of the triangle