Points of Concurrency Objectives: To identify properties of perpendicular bisectors and angle bisectors

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 Cookies Are Best In Milk Chocolate And Ovaltine Perpendicular Bisectors - Circumcenter Angle Bisector - Incenter Medians - Centroid Altitude - Orthocenter

Perpendicular Bisectors circumcenter vertex Construct the perpendicular bisectors of each side of the triangle. **NOT always inside the triangle. **

Perpendicular Bisectors Construct the perpendicular bisectors of each side of the triangle. Perpendicular Bisectors circumcenter vertex

Construct the angle bisectors from each vertex of the triangle. incenter side Construct the angle bisectors from each vertex of the triangle. **ALWAYS inside 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

Construct the angle bisectors from each vertex of the triangle. 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 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.   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.

Construct the median of each side of the triangle. “center of gravity” **Always INSIDE the triangle. ** Medians 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. median vertex midpoint

Construct the median of each side of the triangle. “center of gravity” Medians 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 opposite side. 3. The point of intersection of the three segments is the centroid. median vertex midpoint

Construct the three altitudes of the triangle perpendicular Construct the three altitudes of the triangle **NOT always inside the triangle. ** Obtuse Acute Right

Construct the three altitudes of the triangle perpendicular Construct the three altitudes of the triangle