Geometry concept session 1
2 likes1,367 views
The document discusses concepts from geometry including triangles, polygons, 3D figures and circles. It provides information on the basic properties of triangles such as the sum of interior angles and properties related to sides. It also discusses geometric points within a triangle like the centroid, orthocenter, circumcenter and incenter. It presents problems related to finding lengths, angles and determining triangle types based on given information. The document is a lesson plan on fundamental geometry topics intended to build students' understanding.
Geometry concept session 1
- 1. Alosies George IIM Calcutta director@georgeprep.com +91- 9985-372-371 www.georgeprep.com 1 Geometry Concept Session 1
- 3. What it takes to solve a question from geometry? www.georgeprep.com 3
- 4. www.georgeprep.com 4 TrianglesBasic Properties Geometric Points in a Triangle Congruence and Similarity Equilateral and isosceles triangle Area of a triangle Basic theorems Polygons General properties of Convex and concave polygon properties Regular hexagon and octagon Quadrilaterals • Cyclic quadrilateral • Parallelogram • Rectangle • Rhombus • Square • Trapezium 3 D figures • Cube • Cuboid • Cone • Cylinder • Sphere • Hemisphere • Pyramid • Fulcrum Circle General properties
- 6. Building blocks of Geometry www.georgeprep.com 6 AB C Points, lines and planes Collinear and Coplanar points Collinear Points – points on same line Coplanar points – points on the same plane
- 7. Angles An angle consists of two rays with a common end point (or, initial point). Each ray is a side of the angle. The common endpoint is called the vertex of the angle. www.georgeprep.com 7Building blocks of Geometry
- 8. Classification of angles Acute Right Obtuse Reflex www.georgeprep.com 8Building blocks of Geometry Reflex
- 9. Supplementary angles and complementary angles www.georgeprep.com 9Building blocks of Geometry
- 10. www.georgeprep.com 10 Vertically opposite angles Linear Pair Building blocks of Geometry
- 11. www.georgeprep.com 11 Parallel lines and transversal Building blocks of Geometry
- 12. www.georgeprep.com 12Building blocks of Geometry
- 13. Building blocks of Geometry www.georgeprep.com 13Building blocks of Geometry
- 14. www.georgeprep.com 14 In the figure, AB is parallel to CD, if Angle ABE = 110˚ and angle BED = 25 ˚ find angle CDE ( in degrees) 1. 115 2. 125 3. 135 4. 55 5. None of these A E D B C Answer: Option 3
- 15. www.georgeprep.com 15 AB is parallel to CD. Angle QPB = 105 degrees and Angle QRC = 35 degrees Find the measure of angle PQR ( in degrees) 1. 115 2. 125 3. 120 4. 55 5. None of these A D B C Answer: Option 3 R Q P
- 16. www.georgeprep.com 16 Triangles – Basic Properties
- 17. Triangles Basic Properties Geometric Points in a Triangle Congruence and Similarity Equilateral and isosceles triangles Area of a triangle Basic theorems www.georgeprep.com 17
- 18. Triangles Basic Properties • The sum of three internal angles = 180 • External angles? • Exterior angle is equal to sum of interior opposite Side related properties • Sum of any two sides is greater than the third side • Difference of any two sides is lesser than the third side • Side opposite the greater angle is greater Classification • On the basis of angles • On the basis of sides www.georgeprep.com 18
- 19. Problem www.georgeprep.com 19 The two sides of a triangle are 9cm and 20cm. If the magnitude of the third side is an odd number, what is the sum of all the possible lengths of the third side? 1. 120 2. 160 3. 180 4. None of these
- 20. Problem www.georgeprep.com 20 How many non-congruent triangles can be formed with a perimeter of 9 units with each side having integer unit lengths? 1) 3 2) 2 3) 4 4) None of these
- 21. Sine Rule If two sides of a triangle and an angle opposite one of the two sides is given can you find out the angle opposite the other side? www.georgeprep.com 21 a/Sin A = b/ sin B = c/ Sin C = 2R
- 22. Cosine Rule If three sides of a triangle are given can you find the three angles of the triangle? www.georgeprep.com 22 a2 = b2+c2 – 2bc.cosA b2 = a2+c2 – 2ac.cosB c2 = a2+b2 – 2ca.cosC
- 23. Acute, Right and Obtuse on the basis of sides Can you find out whether a triangle is acute if you know the dimensions of its sides? www.georgeprep.com 23 a2 = b2+c2 – 2bc.cosA b2 = a2+c2 – 2ac.cosB c2 = a2+b2 – 2ab.cosC
- 24. Consider obtuse-angled triangles with sides 8 cm, 15 cm and x cm. If x is an integer then how many such triangles exist? 1. 5 2. 21 3. 10 4. 15 5. 146 - CAT 2008 Problem www.georgeprep.com 24
- 26. Geometric points in a triangle Centroid - medians - 2 triangles of equal area (6 triangles of equal area) - 2:1 ratio - Sum of squares of median/(sum of squares of sides) = 3/4 www.georgeprep.com 26
- 27. Appolonius theorem Geometric points in a triangle Centroid - How can we find out the length of a median if the three sides of a triangle are given? www.georgeprep.com 27 AB2 + AC2 = 2( AD2 + BD2)
- 28. Orthocenter - Altitudes - Position of Orthocenter and the nature of the triangle www.georgeprep.com 28 Geometric points in a triangle
- 29. Orthocenter - If three sides of a triangle are given can you find the foot of the perpendicular from a vertex. www.georgeprep.com 29 Geometric points in a triangle b2 = a2+c2 – 2 BDxBC b2 = a2+c2 + 2 BDxBC
- 30. Circumcenter - Perpendicular bisectors - circumcircle - Position of circumcenter and the nature of the triangle www.georgeprep.com 30Geometric points in a triangle
- 31. Incenter - Angle bisector - Internal angle bisector theorem - Incircle - Angle ADB= 90-(AngleA)/2 - Excenters www.georgeprep.com 31Geometric points in a triangle
- 32. www.georgeprep.com 32Geometric points in a triangle The line that connects the centroid (G), the orthocenter (H), and the circumcenter (C) is called the Euler Line. GH = 2GC. Euler’s Line
- 33. Problem Let PQR be a triangle and O be a point such that areas of triangles POR, POQ and ROQ are equal. If PQR is a scalene triangle which of the following must be true? 1. O lies on the altitude to base QR 2. O lies on the median to base QR 3. O lies on the perpendicular bisector to base QR 4. O lies outside triangle PQR www.georgeprep.com 33