1. Section 3.4 Heidi Frantz, T.J. Murray

2. A. Line segment drawn from triangle vertex to midpoint of opposite side. [Ex. (segment) FB.] B. Line segment drawn from triangle vertex that bisects opposite side. C. Line segment drawn from triangle vertex that divides opposite side into 2 congruent segments (used in proofs).

3. 1. Given: CE congruent EB Conclusion: AE is median to CB Reason: If a segment drawn from a triangle vertex divides the opposite side into 2 congruent segments, then it is a median. 2. Given: BF is median to AC Conclusion: FC congruent FA Reason: If a segment drawn from a triangle vertex is a median, then it divides the opposite side into two congruent segments.

4.  I. Centroid A. Center of gravity of triangle B. Two-thirds of the way from the vertex to the midpoint of the triangle. II. Orthocenter A. Where all 3 altitudes of a triangle intersect B. One of triangle's points of concurrency

5. Altitudes Of Triangles A. Line Segment drawn from triangle vertex perpendicular to opposite side (extended if necessary; proofs). B. Line segment drawn from a triangle vertex that forms right angles with the opposite side (Proofs). C. Line segment drawn from triangle vertex that forms 90 degree angles with the opposite side (Problems).  Every Triangle has 3 altitudes

6. 1. Given: AD is perpendicular to BC Conclusion: AD is alt. to BC Reason: If a segment drawn from a triangle vertex is perpendicular to the opposite side, then it is an altitude.

7. 2. Given: AD is alt. of triangle ABC Conclusion: Angle ADC is a right angle Reason: If a segment drawn from a triangle vertex is an altitude, then it forms right angles with the opposite side.

8.  Given: Triangle ABC is isosceles with base BC  AD is alt. of triangle ABC  Prove: AD is median of triangle ABC  NOTE: DIAGRAM NOT DRAWN TO SCALE 1.

9. 1. Triangle ABC is isos. w/ base BC (Given) 2. AD is alt. of Triangle ABC (Given) 3. AB=AC (If triangle is isos., then sides are congruent.) 4. Angle ADB, ADC are right angles (If a segment drawn from a triangle vertex is an alt., then it forms right angles with the opposite side.) 5. Triangle ADB, ADC are right triangles (If a triangle contains a right angle, then it is a right triangle.) 6.AD=AD (Reflexive) 7. Triangle ADB=ADC (HL, steps 3,5,6) 8. BD=DC (CPCTC) 9. AD is median of ABC (If a segment drawn from a triangle vertex divides the opposite side into congruent segments, then it is a median. Statements are numbered, and reasons are in parenthesis.

10.  Given: AE, FB, and DC are medians. AF=10, AB=45, CE=x+10, EB=2x-10  Find the perimeter of triangle ABC 2.

11. AF=FC (10+10=20) (CE=EB) x+10=2x-10 AC=20, AB=45 20=x x+10=30, 2x-10=30 CB= (30+30) CB=60 AB+AC+CB= (20+45+60) = 125 Therefore, perimeter of triangle ABC is 125 units.

12.  Given: AD is alt. to triangle ABC  Angle BDA=6x  Angle BAD=x  Angle DAC=3x+10  Find: Measure of angle BAC 3.

13. m<BDA=90 degrees, 6x (90=6x) (x=15) m<BAD=x, which is = to 15 degrees. m<DAC=3x+10, which is = to 55 degrees. (m<BAD) + (m<DAC) = (m<BAC) 15 degrees + 55 degrees= 70 degrees Therefore, m<BAC=70 degrees

14.  Given: Angle BDA is a right angle  Conclusion: ________  Reason:______________ _____________________ _____________________ _____________________

15. Conclusion: AD is altitude to triangle ABC Reason: If a line segment drawn from a triangle vertex forms 90 degree angles with the opposite side, then it is an altitude to that side.

16. Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. New ed. Evanston, Illinois: McDougal, Little, and Company, 2004. 131-137. Print. "File:Triangle.Centroid.Median.png." WIKIMEDIA COMMISIONS. 18 January 2009. Web. 17 Jan 2010..