Geom 5point1and2

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Geom 5point1and2

  1. 1. 5.1 & 5.2 Perpendiculars & Bisectors Objectives: - Use properties of perpendicular bisectors - Use properties of angle bisectors to find distances - Use bisectors in a triangle
  2. 2. Perpendicular Bisectors <ul><li>A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector . </li></ul><ul><li>Line CP is a  bisector </li></ul><ul><li>of segment AB </li></ul>P A B C
  3. 3. Perpendicular Bisectors <ul><li>A point is equidistant from two points if its distance from each point is the same. </li></ul><ul><li>In this diagram, if P is the midpoint of AB, then P is equidistant from A and B </li></ul>P A B C
  4. 4. Perpendicular Bisector Theorem <ul><li>If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. </li></ul><ul><li>If CP is the perpendicular bisector of AB, then CA = CB </li></ul>P A B C
  5. 5. Converse of the Perpendicular Bisector Theorem <ul><li>If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of a segment. </li></ul><ul><li>If CA = CB then C lies on the perpendicular bisector of AB. </li></ul>P A B C
  6. 6. Show MN is the perpendicular bisector of ST <ul><li>What segment lengths in the diagram are equal? </li></ul><ul><li>NS and NT, because MN bisects ST </li></ul><ul><li>MS and MT, by the theorem we just learned </li></ul><ul><li>QS and QT, because they are both 12 </li></ul>N T S M Q 12 12
  7. 7. Show MN is the perpendicular bisector of ST <ul><li>Explain why Q is on MN </li></ul><ul><li>QS = QT, so Q is equidistant from S and T. By the converse theorem we just learned, Q is on the perpendicular bisector of ST, which is MN. </li></ul>N T S M Q 12 12
  8. 8. Properties of Angle Bisectors <ul><li>The distance from a point to a line is defined as the length of the perpendicular segment from the point to the line. </li></ul><ul><li>For example, the distance between Q and line m is QP </li></ul>P m Q
  9. 9. Properties of Angle Bisectors <ul><li>When a point is the same distance from one line as it is from another line, then the point is equidistant from the two lines (or rays or segments). </li></ul>
  10. 10. Angle Bisector Theorem <ul><li>If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. </li></ul><ul><li>If m  BAD = m  CAD, then DB = DC </li></ul>A B D C
  11. 11. Converse of Angle Bisector Theorem <ul><li>If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. </li></ul><ul><li>If DB = DC, then m  BAD = m  CAD </li></ul>A B D C
  12. 12. Look at p. 267 Example 3 <ul><li>You are given that B bisects  CAD and that  ACB and  ADB are right angles. What can you say about BC and BD? </li></ul><ul><li>Because BC and BD meet AC and AD at right angles, they are perpendicular segments to the sides of  CAD. </li></ul><ul><li>This implies that their lengths represent the distances from the point B to AC and AD. </li></ul><ul><li>Because point B is on the bisector of  CAD, it is equidistant from the sides of the angle. </li></ul><ul><li>So, BC = BD and you can conclude that segment BC ~= segment BD. </li></ul>
  13. 13. Do p. 267 1-7
  14. 14. Using Perpendicular Bisectors of a Triangle <ul><li>A perpendicular bisector of a triangle is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side. </li></ul>
  15. 15. Concurrent lines <ul><li>When 3 or more lines intersect in the same point, they are called concurrent lines . The point of intersection of the lines is called the point of concurrency . </li></ul>
  16. 16. Concurrent lines <ul><li>The 3 perpendicular bisectors of a triangle are concurrent. The point of concurrency can be inside , on , or outside the triangle. </li></ul><ul><li>The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle . </li></ul>
  17. 17. Concurrency of Perpendicular Bisectors of a Triangle Theorem <ul><li>The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. </li></ul><ul><li>See picture on p. 273 </li></ul>
  18. 18. Useful Stuff! <ul><li>See page 273, Example 1 </li></ul>
  19. 19. Using Angle Bisectors of a Triangle <ul><li>An angle bisector of a triangle is a bisector of an angle of the triangle. </li></ul><ul><li>The 3 angle bisectors are concurrent. </li></ul><ul><li>The point of concurrency is called the incenter of the triangle and always lies inside the triangle. </li></ul>
  20. 20. Concurrency of Angle Bisectors of a Triangle Theorem <ul><li>The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. </li></ul><ul><li>See picture on page 274. </li></ul>
  21. 21. Look at Example 2, p. 274 <ul><li>The angle bisectors of ∆ MNP meet at point L. </li></ul><ul><li>What segments are congruent? </li></ul><ul><li>By the theorem we just learned, the 3 angle bisectors of a triangle intersect at a point that is equidistant from the sides of a triangle. </li></ul><ul><li>So, LR ~= LQ ~= LS </li></ul>
  22. 22. Look at Example 2, p. 274 <ul><li>Find LQ and LR </li></ul><ul><li>Use the Pythagorean Theorem to find LQ in ∆LQM </li></ul><ul><li>(LQ) 2 + (MQ) 2 = (LM) 2 </li></ul><ul><li>(LQ) 2 + 15 2 = 17 2 </li></ul><ul><li>(LQ) 2 + 225 = 289 </li></ul><ul><li>(LQ) 2 = 64 </li></ul><ul><li>LQ = 8 </li></ul><ul><li>Because LQ ~= LR, LR also = 8 </li></ul>
  23. 23. Do p. 275 1-4 <ul><li>Homework: Worksheets </li></ul>

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