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Geom 5point3and4

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Geom 5point3and4

1. 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. 2. Medians <ul><li>A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. </li></ul>P A B C
3. 3. Medians <ul><li>The 3 medians of a triangle are concurrent. </li></ul><ul><li>The point of concurrency is called the centroid of a triangle. The centroid is always inside the triangle. </li></ul>P A B C
4. 4. Concurrency of Medians of a Triangle Theorem <ul><li>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. </li></ul>P A B C
5. 5. Balance <ul><li>A triangular shaped item of uniform thickness and density will balance at the centroid of the triangle. </li></ul>P A B C
6. 6. Using Centroids <ul><li>Q is the centroid of ∆ACP. QB = 5. </li></ul><ul><li>Find AQ and AB. </li></ul><ul><li>RP = (2/3)RT </li></ul><ul><li>PT = RT - RP = </li></ul><ul><li>(1/3) RT </li></ul><ul><li>5 = (1/3)RT </li></ul><ul><li>RT = </li></ul><ul><li>RT = 15 </li></ul><ul><li>RP = (2/3) RT = </li></ul><ul><li>= (2/3)*15 </li></ul><ul><li>= 10 </li></ul>P A B C Q
7. 7. Finding the Centroid of a ∆ <ul><li>Look at the picture at the bottom of p. 280 </li></ul><ul><li>What are the coordinates of N (what is the midpoint of JL? </li></ul><ul><li>5,8 </li></ul><ul><li>Find the distance from K to N. </li></ul><ul><li>6 </li></ul><ul><li>The centroid is (2/3)*6 up from vertex K </li></ul><ul><li>The centroid is 5,6 </li></ul>
8. 8. Altitudes <ul><li>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. </li></ul><ul><li>An altitude can lie inside, on, or outside the triangle </li></ul>
9. 9. Altitudes <ul><li>Every triangle has 3 altitudes. The lines containing the altitudes are concurrent and intersect at a point called the orthocenter of the triangle. </li></ul>
10. 10. Where is the orthocenter in <ul><li>An acute triangle? </li></ul><ul><li>See p. 281 </li></ul><ul><li>A right triangle </li></ul><ul><li>See p. 281 </li></ul><ul><li>An obtuse triangle </li></ul><ul><li>See previous slide & p. 281 </li></ul>
11. 11. Concurrency of Altitudes of a Triangle Theorem <ul><li>The lines containing the altitudes of a triangle are concurrent. </li></ul>
12. 12. Do p. 282 1-7 <ul><li>Homework: worksheets </li></ul>
13. 13. Midsegments <ul><li>A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle </li></ul><ul><li>Make midsegments with paper triangles. </li></ul>
14. 14. Turn to p. 287 <ul><li>And show that midsegment MN is parallel to side JK and half as long. </li></ul><ul><li>Use the Midpoint Formula to find the coordinates of M and N. </li></ul><ul><li>Find the slopes of JK and MN </li></ul><ul><li>Because the slopes are equal, they are parallel. </li></ul><ul><li>Use the Distance Formula to show that MN = √10 and JK = √40 = 2* √10 </li></ul>
15. 15. Midsegment Theorem <ul><li>The segment containing the midpoints of two sides of a triangle is parallel to the third side and is half as long. </li></ul>
16. 16. Do Example 2, p. 288 <ul><li>Do Example 4, p. 289 </li></ul><ul><li>Do Example 5, p. 289 </li></ul><ul><li>Do p. 290 1-11 </li></ul>
17. 17. Homework <ul><li>p. 282 8-12, 18-22 evens </li></ul><ul><li>p. 290 12-14, 24-28 evens </li></ul>