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81. Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.

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82. Let ABC be a right angle in which AB = 6 cm, BC = 8 cm and ∠B = 90°. BD is the perpendicular from B on AC. The circle through B, C, D is drawn. Construct the tangents from A to this circle.

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83. Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of tangents from this point to the circle.

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84. Draw a line segment of length 7.6 cm and divide it in the ratio 3 : 5.

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85. Divide the given line segment AB of length 6 cm internally in the ratio 3 : 4.

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Long Answer Type

86. Divide a line segment 11 cm in the ratio of 2 : 5 (a) internally (b) externally.

(a) Internally : Steps of Construction :

(i)    Draw AB =11 cm.
(ii)    Draw a ray AX making an acute ∠ BAX.
(iii)    Along AX, mark points A₁, A₂, A₃ .........,
A₇. Such that AA_l = A₁A₂ = .......= A₆A₇.
(iv)    Join A₇ B.
(v) Through A₂, draw a line A₂C || A₇B intersecting AB at C.
Thus, point C so obtained is the required point which divides AB internally in the ratio 2
: 5.
(b) Externally : Steps of Construction
(i)    Draw AB = 11 cm.
(ii)    Draw a ray BX making an acute ∠ ABX.
(iii)    Along a ray BX, mark points B₁, B₂,..., B₅, such that BB₁ = B₁B₂ = B₂B₃ =
B₃B₄= B₄B₅.

(a) Internally : Steps of Construction :
(i) Draw AB =11 cm.(ii

(iv) Join B₃A.
(v) Through B₅, draw a line parallel to B₃A intersecting BA produced at C.
Thus, point C so obtained is the required point which divides AB externally in the ratio 2 : 5.
Constructions Based on similar triangle

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Short Answer Type

87. Construct a triangle ABC in which AB = 5 cm, ∠B = 60° and altitude CD = 3 cm. Construct a triangle AQR similar to ΔABC, such that each side of ΔAQR is 1.5 times that of the corresponding side of ΔABC.

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88. Construct a ΔPBC, similar to a given ΔPQR, with QR = 6 cm, PR = PQ = 5 cm, such that each of its side is 5/7th of the corresponding sides of the ΔPQR.

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89. Draw a ΔABC in which BC = 6 cm, AB = 4 cm and AC = 5 cm. Draw a triangle similar to ΔABC with its sides equal to 3/4th of the corresponding sides of ΔABC.

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90. Construct a ΔABC in which BC = 6 cm, m∠BAC = 60° and median through A is 4.5 cm. Construct a ΔA ‘BC’ similar to ΔABC with BC’ = 8 cm.

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