Altitude and Hypotenuse of RightTrianglesTAGUPA, Kristian A.                      P: 555.123.4568 F: 555.123.4567         ...
Lets call this altitude BD.We start with ΔABC.                                                    BA                      ...
This divides the triangle into 2       A    smaller triangles.                                B    A                  D   ...
THEOREM   The    altitude    to   thehypotenuse of a right triangleforms two triangles that aresimilar to the original.   ...
Let’s investigate!            A                         From ΔBDC                         ΔADB                         we ...
The altitude to thehypotenuse is thegeometric mean betweenthe segments intowhich it separates thehypotenuse.              ...
Corollary 1In any right triangle, the altitude tothe hypotenuse is the geometric meanbetween the segments into which itsep...
Let’s investigate! Let’s investigate! A                A           From ΔABC                            ΔACB, we have     ...
AB is the geometricmean of the hypotenuse(AC) and the segmentof the hypotenuseadjacent to AB.               P: 555.123.456...
Let’s investigate!A               A           DB              C   D   B                           P: 555.123.4568 F: 555.1...
Let’s investigate! Let’s investigate!            A                              From ΔBDC                              ΔAB...
BC is the geometricmean of the hypotenuse(AC) and the segmentof the hypotenuseadjacent to BC.               P: 555.123.456...
Let’s investigate!           AB                     DD      C    B            C                             P: 555.123.456...
Corollary 2In any right triangle, each leg is ageometric mean of the hypotenuse andthe segment of the hypotenuse adjacentt...
Let’s investigate further! By Corollary 1,                                            x                        c          ...
Given: Right ΔTRI with altitude RM.a. If IM = 16 and RM = 8. Find TM.By Corollary 1,                                      ...
Given: Right ΔTRI with altitude RM.a. If IT = 25 and TM = 5. Find RM. By Corollary 1,                                     ...
Let’s investigate further! By Corollary 2,                                            x                        c          ...
Let’s investigate further! By Corollary 2,                                            x                        c          ...
Given: Right ΔTRI with altitude RM.a. If IT = 12 and TM = 3. Find TR. By Corollary 2,                                     ...
Practice Exercise:Solve for the unknowns.The given triangle is a righttriangle.                         4                 ...
Practice Exercise:Solve for the unknowns.The given triangle is a righttriangle.                  b                      5 ...
AssignmentSolve for the unknowns.The given triangle is a righttriangle.              4        l               36          ...
THANK YOU!             P: 555.123.4568 F: 555.123.4567             123 West Main Street, New York,             NY 10001   ...
GEOMETRIC MEANDefinitionIf a, b, c are positive numbers, andThen, b is called the geometric meanbetween a and c.          ...
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Geometry

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Geometry

  1. 1. Altitude and Hypotenuse of RightTrianglesTAGUPA, Kristian A. P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  2. 2. Lets call this altitude BD.We start with ΔABC. BA D C P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  3. 3. This divides the triangle into 2 A smaller triangles. B A D CA D CΔADB Band ΔBDC Care formed. ΔABC ΔADB ΔBDC P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  4. 4. THEOREM The altitude to thehypotenuse of a right triangleforms two triangles that aresimilar to the original. P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  5. 5. Let’s investigate! A From ΔBDC ΔADB we haveB D C D B Geometric P: 555.123.4568 F: 555.123.4567 | 123 West Main Street, New York, www.rightcare.com NY 10001 Mean
  6. 6. The altitude to thehypotenuse is thegeometric mean betweenthe segments intowhich it separates thehypotenuse. P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  7. 7. Corollary 1In any right triangle, the altitude tothe hypotenuse is the geometric meanbetween the segments into which itseparates the hypotenuse. P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  8. 8. Let’s investigate! Let’s investigate! A A From ΔABC ΔACB, we have D B C D B P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  9. 9. AB is the geometricmean of the hypotenuse(AC) and the segmentof the hypotenuseadjacent to AB. P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  10. 10. Let’s investigate!A A DB C D B P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  11. 11. Let’s investigate! Let’s investigate! A From ΔBDC ΔABC, we have B D D C B C P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  12. 12. BC is the geometricmean of the hypotenuse(AC) and the segmentof the hypotenuseadjacent to BC. P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  13. 13. Let’s investigate! AB DD C B C P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  14. 14. Corollary 2In any right triangle, each leg is ageometric mean of the hypotenuse andthe segment of the hypotenuse adjacentto the leg. P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  15. 15. Let’s investigate further! By Corollary 1, x c a y h b P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, | www.rightcare.com NY 10001
  16. 16. Given: Right ΔTRI with altitude RM.a. If IM = 16 and RM = 8. Find TM.By Corollary 1, I 16x = 64 M x = 4 T R P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  17. 17. Given: Right ΔTRI with altitude RM.a. If IT = 25 and TM = 5. Find RM. By Corollary 1, I(RM)2 = 100 MRM = 10 T R P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  18. 18. Let’s investigate further! By Corollary 2, x c a y h b P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, | www.rightcare.com NY 10001
  19. 19. Let’s investigate further! By Corollary 2, x c a y h b P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, | www.rightcare.com NY 10001
  20. 20. Given: Right ΔTRI with altitude RM.a. If IT = 12 and TM = 3. Find TR. By Corollary 2, I(TR)2 = 36 MTR = 6 T R P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  21. 21. Practice Exercise:Solve for the unknowns.The given triangle is a righttriangle. 4 a 8 P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  22. 22. Practice Exercise:Solve for the unknowns.The given triangle is a righttriangle. b 5 15 P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  23. 23. AssignmentSolve for the unknowns.The given triangle is a righttriangle. 4 l 36 h g P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  24. 24. THANK YOU! P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com
  25. 25. GEOMETRIC MEANDefinitionIf a, b, c are positive numbers, andThen, b is called the geometric meanbetween a and c. P: 555.123.4568 F: 555.123.4567 123 West Main Street, New York, NY 10001 | www.rightcare.com

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