51. In ΔABC, D and E are the points on AB and AC respectively such that AD × AC = AB × AE. If ∠ADE = ∠ACB + 30° and ∠ABC = 78°, then ∠A = ?
52. In the given figure, if KI = IT and EK = ET, then ∠TEI = . . . . . . . . .
53. If in a triangle ABC, BE and CF are two medians perpendicular to each other and if AB = 19 cm and AC = 22 cm then the length of BC is
54. In a ΔABC, ∠ABC = 2∠CAB, if the side BC is extended to D and ∠ACD = 126°, then ∠CAB is:
55. Two circles of radii 5 cm and 3 cm touch externally, then the ratio in which the direct common tangent to the circles divides externally the line joining the centres of the circles is:
56. In the figure, a circle touches all the four sides of a quadrilateral ABCD whose sides AB = 6.5 cm, BC = 5.4 cm and CD = 5.3 cm. The length of AD is:
57. PQRA is a rectangle, AP = 22 cm, PQ = 8 cm. ΔABC is a triangle whose vertices lie on the sides of PQRA such that BQ = 2 cm and QC = 16 cm. Then the length of the line joining the mid points of the sides AB and BC is.
58. In ΔABC, a line through A cuts the side BC at D such that BD : DC = 4 : 5. If the area of ΔABD = 60 cm2, then the area of ΔADC is:
59. If the perimeter of an isosceles right-angle triangle is 8($$\sqrt 2 $$ + 1) cm, then the length of the hypotenuse of the triangle is:
60. ABCD is a cyclic trapezium whose sides AD and BC are parallel to each other; if ∠ABC = 75° then the measure of ∠BCD is:
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