Question 1:Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse. (i) 7 cm, 24 cm, 25 cm (ii) 3 cm, 8 cm, 6 cm (iii) 50 cm, 80 cm, 100 cm (iv) 13 cm, 12 cm, 5 cm
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Question 2:PQR is a triangle right angled at P and M is a point on QR such that PM ^ QR. Show that PM2 = QM . MR.
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Question 3:In Fig. 6.53, ABD is a triangle right angled at A and AC ^ BD. Show that (i) AB2 = BC . BD (ii) AC2 = BC . DC (iii) AD2 = BD . CD
Question 4:ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2.
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Question 5:ABC is an isosceles triangle with AC = BC. If AB2 = 2 AC2, prove that ABC is a right triangle.
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Question 6:ABC is an equilateral triangle of side 2a. Find each of its altitudes.
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Question 7:Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
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Question 8:In Fig. 6.54, O is a point in the interior of a triangle ABC, OD ^ BC, OE ^ AC and OF ^ AB. Show that (i) OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2, (ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2.
Question 9:A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.
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Question 10:A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?
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Question 11:An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will be the two planes after 1 1 2 hours?
EXERCISE 6.51. Sides of triangles are given below.
Question 12:Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.
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Question 13:D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE2 + BD2 = AB2 + DE2.
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Question 14:The perpendicular from A on side BC of a D ABC intersects BC at D such that DB = 3 CD (see Fig. 6.55). Prove that 2 AB2 = 2 AC2 + BC2.
Question 15:In an equilateral triangle ABC, D is a point on side BC such that BD = 1 3 BC. Prove that 9 AD2 = 7 AB2.
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Question 16:In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.
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Question 17:Tick the correct answer and justify : In D ABC, AB = 6 3 cm, AC = 12 cm and BC = 6 cm. The angle B is : (A) 120° (B) 60° (C) 90° (D) 45°