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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