51. ln a quadrilateral ABCD, E is a point in the interior of the quadrilateral such that DE and CE are the bisectors of ∠D and ∠C, respectively. If ∠B = 82° and ∠DEC = 80°, then ∠A = ?
52. In the following figure, P and Q are centres of two circles. The circles are intersecting at points A and B. PA produced on both the sides meets the circles at C and D. If ∠CPB = 100°, then find the value of x.
53. AC is the diameter of a circle dividing the circle into two semicircles. ED is a chord in one semicircle, such that ED is parallel to AC. B is a point on the circumference of the circle in the other semicircle. ∠CBE = 75°. What is the measure (in degree) of ∠CED?
54. In an equilateral ΔABC, the medians AD, BE and CF intersect to each other at point G. If the area of quadrilateral BDGF is 12√3 cm2, then the side of ΔABC is:
55. In ΔABC, D and E are the midpoints of sides BC and AC, respectively, AD and BE intersect at G at right angle. If AD = 18 cm and BE = 12 cm, then the length of DC (in cm) is:
56. Two circles with centres A and B of radii 5 cm and 3 cm respectively touch each other internally. If the perpendicular bisector of AB meets the bigger circle at P and Q, then the value of PQ is
57. In ΔPQR, ∠Q = 84° and ∠R = 48°, PS ⊥ QR at S and the bisector of ∠P meets QR at T. What is the measure of ∠SPT ?
58. In ΔPQR, ∠P = 90°. S and T are the mid points of sides PR and PQ, respectively. What is the value of $$\frac{{{\text{R}}{{\text{Q}}^2}}}{{{\text{Q}}{{\text{S}}^2} + {\text{R}}{{\text{T}}^2}}} = ?$$
59. In a triangle ABC, AB = AC and the perimeter of ΔABC is 8(2 + √2) cm. If the length of BC is √2 times the length of AB, then find the area of ΔABC.
60. Which of the set of three sides can't form a triangle?
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