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Isosceles Triangles



Question (1): Using Heron's formula find the area of an equilateral triangle whose perimeter is 15 cm.
Answer: Perimeter = 15 cm Each side (a) = 5 cm

Question (2): A triangular field with sides 26 m, 28 m, 30 m has to be leveled. If the cost of leveling is Rs.20 per m². What is the total cost?
Answer: a = 26 m, b = 28 m, c = 30 m = 7 x 6 x 4 x 2 = 336 m² Cost of leveling = 336 x 20 = Rs. 6720

Question (3): The perimeter of a triangle is 42 cm. Two of its sides are 14 cm, 19 cm. Find its area.
Answer: The third side = 42 - (14 + 19) = 9 cm Semi perimeter

Question (4): A wire of length 32 cm is to be bent in the form of an isosceles triangle of base 12 cm. What is the area of the triangle so formed?
Answer: Let each of the equal sides be 'x' cm 2x + 12 = 32 x = 10 cm = 48 cm²

Question (5): The sides of a triangle are in the ratio 12:17:25. Its perimeter is 108 m. Find its area.
Answer: Sum of all the sides = 108 m The ratio of sides = 12: 17: 25 12 + 17 + 25 = 54 = 3 x 6 x 5 x 4 = 360 m²

Question (6): A farmer has a triangular plot of land on which he wants to grow carrots and Radish. He puts a separator along the line AD, D being the mid-point of BC. He grows carrots on one part and Radish in the other. What is the area in hectares used for cultivation of carrots?
Answer: First let us find area of ABC = 33600 m² = 3.36 hectares AD is the median of ABC and hence divided the triangle into two parts of equal area. Area used for cultivation of carrots

Question (7): The sides of a triangle are 16 cm, 30 cm, 34 cm. Find its area using Hero's formula. Also find the length of the altitude drawn on the longest side.
Answer: The longest side = 34 cm Let 'h' be the altitude drawn on the longest side. 17h = 240

Question (8): This is a sign board. The area between the two triangles has to be pained red. The cost of painting is Rs.3.50/cm².What is the total cost?
Answer: Area of PQR : = 300 x 3.87 = 1161 cm² Area of ABC : = 290.25 cm² area to be painted = 1161 - 290.25 = 870.75 cm² Cost of painting = 870. 75 x 350 = Rs. 3047.63

Question (9): ABCD is a kite made of colour paper. How many metres of colour paper is required if the width of the paper is 40 cm.
Answer: Area of ABC: = 1200 cm² Area of DBC = 4512 cm²Area of the kite = 1200 + 4512 = 5712 cm² Width of the paper = 40 cm = 1.428 m

Question (10): The perimeter of rhombus is 48 cm. One of the diagonals is 16 cm. What is its area.
Answer: Perimeter = 48 cm Area of ACD : Area of the rhombus = 2 x area of the triangle

Question (11): Find the area of the quadrilateral ABCD in which AB = 14 cm, BC = 24 cm, CD = 24 cm, AD = 18 cm, AC = 30 cm.
Answer: In ABC, = 40 x 4.12 = 164.8 cm² in ACD, Area of the quadrilateral = 164.8 + 216 = 380.8 cm²

Question (12): Find the area of a quadrilateral ABCD whose sides are 9m, 40 m, 28 m and 15 m respectively and the angle between the first two sides is a right angle.
Answer: AC² = AB² + BC² = 81 + 1600 = 1681 Therefore AC= 41 m Area of ABC = 126 m² Area of the quadrilateral = 180 + 126 = 306 m²

Question (13): A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m. The non-parallel sides are 14 m and 13 m. Find the area of the field.
Answer: Let us draw BE \|\| AD Then ABED is a parallelogram BE = 13 m DE = 10 m EC = 15 m For BEC, Height of trapezium = 11.2 m Area of field = 196 m²

Question (14): The perimeter of a right triangle is144 cm and its hypotenuse measures 65 cm. Find the length of the other sides and calculate its area. Verify the result using Hero's formula.
Answer: a + b + 65 = 144 a + b = 79 ... (1) a² + b² = 65² (Pythagoras theorem) = 4225 Squaring (1) (a + b)² = 79² = 6241 2ab = 6241 - 4225 = 2016 (a - b)² = a² + b² - 2ab = 4225 - 2016 = 2209 a + b = 79 a - b = 47 2a = 126 a = 63 b = 16 Hero's Formula = 504 cm² Hence verified.

Question (15): At the centre of a floor there is a pattern made of 8 triangular tiles of the same size. The shaded portions are made of green tiles and the rest are while. The sides of the tiles are 19.5 cm, 21 cm, 22.5 cm respectively. The cost of the green tiles is Rs.30/100 cm² and the cost of the white tiles is Rs.25/100 cm² and the cost of the white tiles is Rs.25/100 cm². What is the cost of making the pattern?
Answer: The areas of all the triangles are equal. The semi perimeter of one = 31.5 = 108 cm² Area of 4 green tiles = 108 x 4 = 432 cm² The cost of green tiles = 4.32 x 30 = Rs. 129.60 \|\|\|^lyCost of white tiles = 4.32 x 25 = Rs. 108 Total cost = Rs. 237.60

Question (16): Two adjacent sides of a parallelogram measure 15 cm and 10.5 cm. One of its diagonal measures 19.5 cm. What is the area of the parallelogram. Find the length of the altitude drawn to the longer side.
Answer: Semi perimeter of ABC Area of the parallelogram Area of parallelogram = b x h = 15 x DP Therefore

Question (17): A trapezium has parallel sides of lengths 9 cm and 5 cm. The other two sides are of length 2.5 cm. Find its area
Answer: BE is drawn \|\| AD ADEB is a parallelogram DE = 5 cm EC = 4 cm BE= 2.5 cm Semi perimeter of BEC

Question (18): A field is in the shape of a rhombus with each side 75 m and one of its diagonals is 42 m. If a man working in the field can level 63 m² of the field in one day and there are 4 men working how many days will they take to level the field?
Answer: Semi perimeter of ABD = 4 x 6 x 21 x 3 = 1512 m² Area of field = 1512 x 2 = 3024 m² Area of field which 4 men can level in 1 day = 63 x 4 = 252 m² Number of days they need to level the field

Question (19): This is a model of an aeroplane, made of colour paper. Find the total area of the paper used.
Answer: ABC is an isosceles triangle with sides 5 cm, 5 cm, 1 cm Part II is the rectangle BCGH with sides 1 cm and 6.5 cm long. Area of II = 1 x 6.5 = 6.5 cm² Part III HGIJ is a trapezium Area of equilateral GKI IV and V are right angled triangles each with area Therefore total area = 2.3 + 6.5 + 1.3 + 4.5 + 4.5 = 19.1 cm²

Question (20): An umbrella is made by stitching 10 triangular pieces of cloth of two different colours, each piece measuring 20 cm, 50 cm, 50 cm. How much cloth of each colour is used in the umbrella?
Answer: Semi-perimeter of each triangle Area of 1 triangle Area of cloth of each colour = 5 x 490 cm² = 2450 cm²