1. 5-3 Concurrent Lines, Medians, and Altitudes
1/9/17
5-3 Concurrent Lines, Medians, and Altitudes
Objective: To identify properties of perpendicular bisectors, angle bisectors, medians, and altitudes of a triangle.
THEOREM 5-6
The perpendicular bisectors of the sides of a triangle
are concurrent at a point equidistant from the vertices.
THEOREM 5-7
The bisectors of the angles of a triangle are concurrent
at a point equidistant from the sides.

2. CONCURRENT: three or more lines that intersect at one point are “concurrent lines”.
POINT OF CONCURRENCY: The point at which they intersect.

3. Diagram for Theorem 5-6: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.
CIRCUMCENTER of the triangle is the point of concurrency of the perpendicular bisectors of a triangle. The circle is circumscribed about the triangle.
QC = SC = RC S C
Q R .
(But they are NOT angle bisectors)

4. Ex: Find the center of the circle that you can circumscribe about OPS.
Two perpendicular bisectors of sides of OPS are x = 2 and y = 3. These lines intersect at (2,3). This point is the center of the circle P
1) Find the center of the circle that you can circumscribe about the triangle with vertices (0,0), (-8,0), and (0,6). Draw it first. x O S y
Circumcenter of the triangle and center of the circle is (-4, 3).

5. Diagram for Theorem 5-7: The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.
INCENTER OF THE TRIANGLE: the point of concurrency of the angle bisectors of a triangle
The circle is inscribed in the triangle.
I is the incenter of the triangle. It is equidistant from the sides (XI = YI = ZI).

6. MEDIAN OF A TRIANGLE: a segment whose endpoints are a vertex and the midpoint of the opposite side. This is different than an angle bisector!!!
THEOREM 5-8
The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.
DC = 2 DJ H
EC = 2 EG C
FC = 2 FH J
necessarily
NOTE: The medians are NOT angle bisectors! For example, Angle DFH ≠ Angle HFE.

7. Ex: D is the centroid of ABC and DE = 6. Find BE. Since D is
CENTROID: the point of concurrency of the medians (also called the center of gravity).
Ex: D is the centroid of ABC and DE = 6. Find BE. Since D is
a centroid, BD = 2 DE and BE = BD + DE
BD = 2 * DE
BD = 2 * 6
BD = 12, and BE = 18 12 6

8. ACUTE TRIANGLE RIGHT TRIANGLE OBTUSE TRIANGLE
ALTITUDE OF A TRIANGLE: the perpendicular segment from a vertex to the opposite side.
ACUTE TRIANGLE RIGHT TRIANGLE OBTUSE TRIANGLE
Altitude is inside Altitude is a side Altitude is outside
Ex: Is ST a median, an altitude, both, or neither? Explain.
ST VU. ST is an altitude of VSU S W
V T U
1) Is UW a median an altitude, both, or neither? Explain.
Since SW = WV, UW is a median.

9. THEOREM 5-9
The altitudes of a triangle are concurrent at the orthocenter.

10. To find the centroid of a triangle, you need to draw at least ______median(s).
FGH has vertices F (-1,2), G (9,2) and H (9,0). Find the center of
the circle that circumscribes FGH. V X
Y L M S
P Z
Identify all medians and altitudes drawn in PSV.
If SY = 15, find SM and MY
If MX = 14, find PM and PX 2
(4, 1) 14
Med. SY
Med. PX
Alt. VZ
SM = 10, MY = 5
PM = 28, PX = 42

11. SUMMARY Assignment: Page 259 #1, 2, 5, 8, 9, 11 – 16, 19 – 22
Lines Concurrent at Circle Bisectors Circumcenter Outside the Δ Bisectors Incenter Inside the Δ Medians Centroid Altitudes Orthocenter
Assignment: Page 259 #1, 2, 5, 8, 9, 11 – 16, 19 – 22