- The three medians of a triangle intersect at the centroid, which is located 2/3 of the distance from each vertex to the midpoint of the opposite side. - The three altitudes of a triangle are concurrent at the orthocenter, which can lie inside, on, or outside the triangle depending on whether the triangle is acute, right, or obtuse. - A midsegment connects the midpoints of two sides of a triangle. The midsegment is parallel to the third side and is half as long based on the Midsegment Theorem.

1. 5.3 & 5.4 Medians, Altitudes & Midsegments Objectives: - Use properties of medians of a triangle - Use properties of altitudes of a triangle - Use properties of midsegments of a triangle

2. Medians A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. P A B C

3. Medians The 3 medians of a triangle are concurrent. The point of concurrency is called the centroid of a triangle. The centroid is always inside the triangle. P A B C

4. Concurrency of Medians of a Triangle Theorem The medians of a triangle intersect at a point that is 2/3 of the distance from each vertex to the midpoint of the opposite side. P A B C

5. Balance A triangular shaped item of uniform thickness and density will balance at the centroid of the triangle. P A B C

6. Using Centroids Q is the centroid of ∆ACP. QB = 5. Find AQ and AB. RP = (2/3)RT PT = RT - RP = (1/3) RT 5 = (1/3)RT RT = RT = 15 RP = (2/3) RT = = (2/3)*15 = 10 P A B C Q

7. Finding the Centroid of a ∆ Look at the picture at the bottom of p. 280 What are the coordinates of N (what is the midpoint of JL? 5,8 Find the distance from K to N. 6 The centroid is (2/3)*6 up from vertex K The centroid is 5,6

8. Altitudes An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. An altitude can lie inside, on, or outside the triangle

9. Altitudes Every triangle has 3 altitudes. The lines containing the altitudes are concurrent and intersect at a point called the orthocenter of the triangle.

10. Where is the orthocenter in An acute triangle? See p. 281 A right triangle See p. 281 An obtuse triangle See previous slide & p. 281

11. Concurrency of Altitudes of a Triangle Theorem The lines containing the altitudes of a triangle are concurrent.

12. Do p. 282 1-7 Homework: worksheets

13. Midsegments A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle Make midsegments with paper triangles.

14. Turn to p. 287 And show that midsegment MN is parallel to side JK and half as long. Use the Midpoint Formula to find the coordinates of M and N. Find the slopes of JK and MN Because the slopes are equal, they are parallel. Use the Distance Formula to show that MN = √10 and JK = √40 = 2* √10

15. Midsegment Theorem The segment containing the midpoints of two sides of a triangle is parallel to the third side and is half as long.

16. Do Example 2, p. 288 Do Example 4, p. 289 Do Example 5, p. 289 Do p. 290 1-11

17. Homework p. 282 8-12, 18-22 evens p. 290 12-14, 24-28 evens