Chapter 6-4 Notes

Definitions

• Median

• The line segment that joins a and the of the opposite side

of a triangle.

• Does not necessarily

• Is not necessarily

vertex midpoint

bisect the angle

perpendicular

Examples

• Example 1

Draw the median LO for ΔLMN below.

Examples

• Example 2

Find the other two medians for ΔLMN.

Examples

• Example 3

Find the equation of the median from B to the midpoint of AC for the triangle below.

(−𝟔+𝟔

𝟐,−𝟒+ −𝟒

𝟐)

(𝟎

𝟐,−𝟖

𝟐)

(𝟎, −𝟒)

−𝟒−𝟒

𝟎−−𝟐𝒚 − 𝟒 = −𝟒(𝒙 − −𝟐)

-4

−𝟖

𝟐

𝒚 − 𝟒 = −𝟒(𝒙 + 𝟐)

𝒚 − 𝟒 = −𝟒𝒙 − 𝟖

𝒚 = −𝟒𝒙 − 𝟒

Examples

• Example 3.5

Find the equation of the median from A to the midpoint of BC for the triangle in Example 3.

(−𝟐+𝟔

𝟐,𝟒+ −𝟒

𝟐)

(𝟒

𝟐,𝟎

𝟐)

(𝟐, 𝟎)

𝟎−−𝟒

𝟐−−𝟔 𝒚 − 𝟎 =𝟏

𝟐(𝒙 − 𝟐)

𝟏

𝟐

𝟒

𝟖 𝒚 =𝟏

𝟐𝒙 − 𝟏

Definitions

• Centroid

• The point of concurrency for the medians of a triangle

Theorems

• Concurrency of Medians Theorem

• The centroid is of the distance from the vertices to the midpoint

of the opposite side.

• The centroid is the of a triangle

two-thirds

balancing point

Note

• Using the triangle below, where G is the centroid, we can conclude:

𝑨𝑮 =𝟐

𝟑𝑨𝑫

𝑪𝑮 =𝟐

𝟑𝑪𝑭

𝑬𝑮 =𝟐

𝟑𝐁𝐄

𝑫𝑮 =𝟏

𝟑𝑨𝑫

𝑭𝑮 =𝟏

𝟑𝑪𝑭

𝑩𝑮 =𝟏

𝟑𝑩𝑬

𝑫𝑮 =𝟏

𝟐𝑨𝑮

𝑭𝑮 =𝟏

𝟐𝑪𝑮

𝑩𝑮 =𝟏

𝟐𝑬𝑮

Examples

• Example 4

I, K, and M are midpoints of the sides of ΔHJL. If JM = 18, find JN and NM. If HN = 14, find NK and HK.

𝑱𝑵 = 𝟏𝟐

𝑵𝑴 = 𝟔

𝑵𝑲 = 𝟕

𝑯𝑲 = 𝟐𝟏

Examples

• Example 5

H is the centroid of ΔABC and DC = 5y – 16. Find x and y.

3x + 6 = 2(2x – 1)

3x + 6 = 4x – 26 = x – 28 = x

𝟐

𝟑(5y – 16) = 2y + 8

2(5y – 16) = 3(2y + 8)

10y – 32 = 6y + 24

4y – 32 = 24

4y = 56y = 14

Definitions

• Altitude

• A line segment from a vertex and to the opposite side

• Also know as the of a triangle

perpendicular

height

Definitions

• Orthocenter

• The point of concurrency for the of a triangle

• In an acute triangle, the orthocenter is the triangle

• In a right triangle, the orthocenter is the of the right angle.

• In an obtuse triangle, the orthocenter is the triangle.

• The orthocenter has no special proeprties

altitudes

inside

vertex

outside