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Olympiad 3

2. cos A + cos B + cos C

2. cos A + cos B + cos C = 4 sin A 2 sin B 2 sin C 2 3. tan A + tan B + tan C = tan A tan B tan C 4. tan A 2 tan B 2 + tan A 2 tan C 2 + tan B 2 tan C 2 = 1 Exercise 22. ABCD is a quadrilateral whose diagonals intersect at O. If ∠AOB = 30 ◦ , AC = 24 cm and BD = 22 cm, find the area of ABCD. B A 30 ◦ O D C Exercise 23. In the figure, ABCDEF is a regular hexagon with area equal to 3 √ 3 cm 2 . Find the area of the square P QRS. Q C B P D A R E F S Exercise 24. ABCD is a square, AEF is a n isosceles triangle, and ∠EAF = 30 ◦ . Points E and F lie on BC and CD respectively. The area of △AEF is 1. Find the area of ABCD. D 30 ◦ F C E A B Exercise 25. The perimeters of an equilateral triangle and a regular hexagon are equal. If the area of the triangle is 5 square units. Find the area of the hexagon in square units. 6 Score:

Exercise 26. △ABC is a rightangled triangle with AB⊥BC and AB = BC. D is a point such that AD⊥BD with AD = 5 and BD = 8, find the value of the area of △BCD. A 5 D 8 C A Exercise 27. B Given ∠ABC = 90 ◦ , AC = BC = 14 cm and CE = CF = 6 cm. If CD = x find the value of x. E D C F B Exercise 28. In △ABC, we have AB = 1 and AC = 2. Side BC and the median from A to BC have the same length. What is BC? Exercise 29. In the figure, ∠ABC = 30 ◦ and AC = 45 cm. Find the radius of the circumcircle of △ABC in cm. A 45 cm B 30 ◦ C Exercise 30. The sines of the angles of a triangle are in the ratio 3 : 4 : 5. If A is the smallest interior angle of the triangle and cos A = x , find the value 5 of x. Exercise 31. In the figure, AB = AC = 6 cm and BC = 9.6 cm. Find the diameter of the circumcircle of △ABC in cm. 7 Score:

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