Boards Concepts Booster Class 11 Solution and Properties of Triangle

1. BOARDS CONCEPTS BOOSTER SOLUTION AND PROPERTIES OF TRIANGLE  Download Doubtnut Today Ques No. Question CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 1. INTRODUCTION 1 1. Sine formula proof :Sine Rule  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 2. THE LAWS OF SINESORSINERULE SINESORSINERULE 2 3 1. Problem on sine rule Type:-1(i) In a ΔABC ; If a = 2; b = 3 and sinA = 2 a = 2;b = 3 and 2 sinA = 3 ;find ∠B (ii) In a ΔABC; the angle of a triangle are in AP ; It is being given that b:c =√3:√2  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 2. THE LAWS OF SINESORSINERULE SINESORSINERULE 2. An object is observed from three points A, B,C in the same horizontal line passing through the base of the object. The angle of elevation at B is twice and at C is thrice than that at A. If AB = a,BC = b prove that the height of the object is ( ) 3 a 2b√(a + b)(3b - a) h =  Click to LEARN this concept/topic on Doubtnut

2. CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 3. THE LAW OF COSINES 4 1. The law of cosines proof :  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 4. PROJECTION FORMULAE 5 1. In ΔABC prove that (i)a = bcosC + ccosB (ii)b = ccosA + acosC (iii) c = acosB + bcosA  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 5. PROJECTION FORMULA ( ) ( ) ( ) 6 A 2 B 2 C 2 1. Half Angle Formula of sin ,sin , sin  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 6. HALF-ANGLE FORMULAS ( ) ( ) ( ) 7 A 2 B 2 C 2 1. Half Angle Formula of cos , cos ,cos  Click to LEARN this concept/topic on Doubtnut

3. CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 6. HALF-ANGLE FORMULAS ( ) ( ) ( ) 8 A 2 B 2 C 2 2. Half Angle Formula of tan ,tan ,tan  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 7. NAPIER'S ANALOGY ( ) ( cot ) ( ) ( ) ( cot ) ( ) B - C 2 b - c b + c A 2 A - B 2 a - b a + b C 2 1. In any ΔABC (i)tan (ii) tan (iii) = = 9 ( ) ( cot ) ( ) C - A 2 c - a c + a B 2 tan =  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 8. AREA OF TRIANGLE 10 1. Verify area of triangle is (i) 1 2bcsinA (ii) Herons Formula  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF

4. TRIANGLE 8. AREA OF TRIANGLE 11 2. Verify area of triangle is (i) abc/4R (ii) 2 R^2 SinA sinB sinC  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 12 1. (1)What is Circumcenter and Circumcircle of Triangle (2) What are the Position of Circumcentre wrt type of triangle ?  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 13 2. What is the Distance of Circumcenter from vertex of triangle ?  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 14 3. Formulas for Circumradius  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF

5. TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 15 4. Explain Incenter and Incircle. Show incenter is Intersection point of angle bisector  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 16 5. State Angle Bisector Theorem and prove it .  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 17 6. (i) Find the Distance of incenter from vertex (ii)Find the Length of tangent from Vertices to incircle  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 18 7. (i)Find the length of Angle Bisector  Click to LEARN this concept/topic on Doubtnut

6. CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE ( ) ( ) ( ) ( ) 19 8. Derive the Formula of inradius (i)Δ A 2 A 2 B 2 C 2 s(ii)(s - a)tan (iii)4Rsin sin sin  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 20 9. Distance Between Circumcenter and incenter  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 21 10. (i)Explain Orthocenter with diagram and position of orthocenter in acute triangle; right triangle and obtuse triangle .  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 22

7. 11. Distance of Orthocenter from sides and vertices  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 23 12. Distance between circumcenter and orthocenter ?  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 24 13. What is pedal triangle In acute angle triangle (orthocenter of ABC is Incenter of Pedal Triangle.)?  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 25 14. Side of pedal triangle and Circum-Radius of pedal Triangle  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF

8. TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 26 15. (i)Centroid of a triangle (ii) Length of Median  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 27 16. State Apollonius Theorem and Centroid divides circumcentre and orthocentre in 1:2  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 28 17. Escribed Circle  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 18. Find the Radius of escribed circle 29

9. ( ) ( ) ( ) ( ) Δ A 2 A 2 B 2 C 2 (i)r1= = stan = 4Rsin cos cos s - a  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 30 19. Prove that r1+ r2+ r3- r = 4R  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 31 r R 20. Prove that cosA + cosB + cosC = 1 +  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 32 21. If in a triangle r1= r2+ r3+ r; Prove that triangle is right angled.  Click to LEARN this concept/topic on Doubtnut

10. CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 9. DIFFERENT CIRCLES AND CENTERS CONNECTED WITH TRIANGLE 33 22. Prove that acosA + bcosB + ccosC r R = a + b + c  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 10. SOME IMPORTANT RESULTS AND THEIR APPLICATIONS ) ( ) ( )( ) ( )( ) 34 B - C 2 tan(C - A) 2 tan(A - B) 2 ( 1. Prove that r + r1tan = r + r2 = r + r3 = 0  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 10. SOME IMPORTANT RESULTS AND THEIR APPLICATIONS 2. Given a triangle ABC with sides a = 7; b = 8 and c = 5 Find the value of expressions ( ( ) 35 ( ) ( )) A 2 B 2 C 2 (sinA + sinB + sinC) cot + cot + cot  Click to LEARN this concept/topic on Doubtnut

11. CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 10. SOME IMPORTANT RESULTS AND THEIR APPLICATIONS 36 3. State Geometrical relation of Excenters  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 10. SOME IMPORTANT RESULTS AND THEIR APPLICATIONS 37 4. Distance between Incenter and excenter  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 10. SOME IMPORTANT RESULTS AND THEIR APPLICATIONS 38 5. Distance between Excenters  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 11. MISCELLANEOUS TOPICS 39 1. M-N THEOREM  Click to LEARN this concept/topic on Doubtnut

12. CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 11. MISCELLANEOUS TOPICS 40 2. Area of the Quadrilateral  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 11. MISCELLANEOUS TOPICS 41 3. Cyclic Quadrilateral and its properties  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 11. MISCELLANEOUS TOPICS 42 4. Regular polygon and its area  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 12. SOLUTION OF TRIANGLE 1. Solution of Triangle in different cases :- (1) When b < csinB (2) b = csinB and B is an acute angle (3)b > csinB; b < c and B is an acute angle (4) b > csinB; c < b;and B is an acute angle 43  Click to LEARN this concept/topic on Doubtnut

13. CONCEPT FOR BOARDS || Chapter SOLUTION AND PROPERTIES OF TRIANGLE 12. SOLUTION OF TRIANGLE 2. Solution of Triangle in different cases :- (5)When b > csinB; c < b; and B is an obtuse angle (6) when b > c; B = 90∘ (7) b ≤ c and B = 90∘ 44  Click to LEARN this concept/topic on Doubtnut  Download Doubtnut to Ask Any Math Question By just a click  Get A Video Solution For Free in Seconds  Doubtnut Has More Than 1 Lakh Video Solutions  Free Video Solutions of NCERT, RD Sharma, RS Aggarwal, Cengage (G.Tewani), Resonance DPP, Allen, Bansal, FIITJEE, Akash, Narayana, VidyaMandir  Download Doubtnut Today