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(i) 7 cm, 24 cm, 25 cm 7 2 = 49 24 2 = 576 and 25 2 = 625 We see that 7 2 + 24 2 = 25 2 ∴ The given triangle is right angled Hypotenuse = 25 cm. (ii) 3 cm, 8 cm, 6 cm 3 2 = 9 8 2 = 64 6 2 = 36 ∵ 3 2 + 6 2 ≠ 8 2 ∴ The given triangle is not right angled. (iii) 50 cm, 80 cm, 100 cm 50 2 = 2500 80 2 = 6400 100 2 = 10000 ∵ 50 2 + 80 2 ≠ 100 2 ∴ The given triangle is not right angled. (iv) 13 cm, 12 cm, 5 cm 13 2 = 169 12 2 = 144 5 2 = 25 ∵ 5 2 + 12 2 ≠ 13 2 ∴ The given triangle is right angled Hypotenuse = 13 cm.

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Triangles

Short Answer Type

211. Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.

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212. ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is
(A) 2 : 1 (B) 1 : 2 (C) 4 : 1 (D) 1 : 4.

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213.

Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio
(a) 2 : 3 (b) 4 : 9
(c) 81 : 16 (d) 16 : 81.

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Long Answer Type

214. Sides of some triangles are given below. Determine which of them are right triangles. In case of a right angle, 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.

(i) 7 cm, 24 cm, 25 cm
7² = 49
24² = 576
and    25² = 625
We see that
7² + 24² = 25²∴ The given triangle is right angled
Hypotenuse = 25 cm.
(ii) 3 cm, 8 cm, 6 cm
3² = 9
8² = 64
6² = 36
∵    3² + 6² ≠ 8²∴ The given triangle is not right angled.
(iii)    50 cm, 80 cm, 100 cm
50² = 2500
80² = 6400
100² = 10000
∵    50² + 80² ≠ 100²∴ The given triangle is not right angled.
(iv)    13 cm, 12 cm, 5 cm
13² = 169
12² = 144
5² = 25
∵    5² + 12² ≠ 13²∴ The given triangle is right angled
Hypotenuse = 13 cm.