Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Geom 5point3and4

937 views

Published on

Published in: Business, Real Estate
  • Be the first to comment

  • Be the first to like this

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>

×