Geometry Concept session 2
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This document discusses geometry concepts related to triangles, including: - Congruency and similarity conditions for triangles - Properties of right triangles like Pythagorean theorem, circumradius, inradius, and area formula - Special right triangles like 45-45-90 and 30-60-90 triangles - Properties and formulas for equilateral, isosceles, and general triangles like Heron's formula - Worked problems are provided as examples.
![Special Triangles – Isosceles Triangle
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21
• Exactly two sides are equal
• Exactly two angles are equal
• AD = median, angle bisector, altitude and
perpendicular bisector
• Area = [b√(4a2 –b2)]/4](https://image.slidesharecdn.com/geometryconceptsession2-151010110635-lva1-app6892/85/Geometry-Concept-session-2-19-320.jpg)
Geometry Concept session 2
- 1. Alosies George IIM Calcutta director@georgeprep.com +91- 9985-372-371 www.georgeprep.com 1 Geometry Concept Session 2
- 2. www.georgeprep.com 2 Triangles – Congruency and Similarity
- 3. Congruency www.georgeprep.com 3 - Corresponding sides and corresponding angles are equal - Congruency conditions - SSS - SAS - AAS - ASA - RHS - Properties of two congruent triangles
- 4. Similarity www.georgeprep.com 4 - Corresponding sides proportional; corresponding angles are equal - Any congruency condition is also a similarity condition(the vice versa is not true) - Properties of two similar triangles
- 5. www.georgeprep.com 5 ABC is a right angled triangle and E is a point chosen on AC such that the rectangle formed inside ABC by dropping the perpendiculars ED and EF from E has the maximum area possible. What is the area of the triangle EFC in that case, if AB = 12cm and AC = 13cm? 1. 12 2. 12.5 3. 7.5 4. 10 5. None of these Problems
- 6. www.georgeprep.com 6 ABCD is a trapezium with the two parallel sides measuring 15cm and 9cm respectively. A line segment PQ, parallel to the parallel sides, is drawn such that P is on AB and Q is on CD and PQ divides the trapezium into two figures of equal area. The length of the line segment PQ is 1. 168 2. 153 3. 8 4. None of these Problems
- 7. www.georgeprep.com 7 Special Triangles and their Properties
- 8. Special Triangles – Right Angled Triangle Pythagoras theorem Pythagoras triplets www.georgeprep.com 8
- 10. Special Triangles – Right Angled Triangle Circumradius Inradius Area www.georgeprep.com 11 R = ℎ 2 A = 𝑎 ∗ 𝑏 2 r = 𝑎+𝑏−ℎ 2 = 𝑎𝑏 𝑎+𝑏−𝑐 h a b
- 11. Special Triangles – Right Angled Triangle –(Altitude-on- Hypotenuse Theorem) www.georgeprep.com 12 In a right triangle, the altitude that’s perpendicular to the hypotenuse creates two smaller right triangles that are both similar to the original right triangle.
- 13. Special Triangles – Right Angled Triangle Special Right Angled triangles www.georgeprep.com 14 90-60-30 triangle 90-45-45 triangle Side Ratios 2: √3:1 √2 : 1: 1
- 14. www.georgeprep.com 15 - CAT 2007 Answer : Option 1 Special Right Triangles ( 90-45-45)
- 15. www.georgeprep.com 16 Answer : Option 1 Problems
- 16. Special Triangles – Equilateral Triangle www.georgeprep.com 18 In terms of side, a In terms of altitude, h Side - 2h/ √3 Altitude √3a/2 - Circumradius a/√3 2/3 h Inradius a/2√3 1/3 h Area √3a2 /4 h2/√3
- 17. Special Triangles – Equilateral Triangle www.georgeprep.com 19 AD = median, perpendicular bisector, Angle Bisector and altitude E = centroid, incenter, orthocenter and circumcenter For a given perimeter, Equilateral Triangle has the largest Area For a given area, the equilateral triangle has the least perimeter.
- 19. Special Triangles – Isosceles Triangle www.georgeprep.com 21 • Exactly two sides are equal • Exactly two angles are equal • AD = median, angle bisector, altitude and perpendicular bisector • Area = [b√(4a2 –b2)]/4
- 20. Area of Triangles www.georgeprep.com 22 ½ breadth x height Heron’s formula ab sin(included angle)/2 abc/4R (where R is circumradius) rs ( where r is inradius and s is semiperimeter)
- 21. www.georgeprep.com 23Problems Answer : Option B PQ and PR are two sides of a triangle that make an angle of 120 degrees between them. If the angle bisector of angle QPR is of length 4cm, what is the harmonic mean of the lengths of PQ and PR? 1. 4 2. 8 3. 4/ 3 4. None of these