CK-12 Geometry - Second Edition

5.4. Medians and Altitudes in Triangles http://www.ck12.org

In addition to these ratios,Gis also the balance point of 4 ACE. This means that the triangle will balance when
placed on a pencil (#3 in Investigation 5-3) at this point.

Example 4:I,K, andMare midpoints of the sides of 4 HJL.

a) IfJM=18, findJNandNM.

b) IfHN=14, findNKandHK.

Solution:

a)JNis two-thirds ofJM. So,JN=^23 · 18 =12.NMis either half of 12, a third of 18 or 18−12.NM=6.

b)HNis two-thirds ofHK. So, 14=^23 ·HKandHK= 14 ·^32 =21.NKis a third of 21, half of 14, or 21−14.
NK=7.

Example 5:Algebra ConnectionHis the centroid of 4 ABCandDC= 5 y−16. Findxandy.

Solution:HFis half ofBH. Use this information to solve forx. Fory,HCis two-thirds ofDC. Set up an equation
for both.

1

2

BH=HForBH= 2 HF HC=

2

3

DCor

3

2

HC=DC

3 x+ 6 = 2 ( 2 x− 1 )

3

2

( 2 y+ 8 ) = 5 y− 16

3 x+ 6 = 4 x− 2 3 y+ 12 = 5 y− 16

8 =x 28 = 2 y