Question 12 of 16intermediate🔧 ApplyNumerical5 marks

Let a, b and c denote the length of the sides of a right triangle, with c being the length of the hypotenuse. Find the missing sidelength in each of the following cases:

(i)

a = 5, b = 7

Answer

c = 52+72\sqrt{5² + 7²} = 25+49\sqrt{25 + 49} = 74\sqrt{74} ≈\approx 8.60 units

(ii)

a = 8, b = 12

Answer

c = 82+122\sqrt{8² + 12²} = 64+144\sqrt{64 + 144} = 208\sqrt{208} = 413\sqrt{13} ≈\approx 14.42 units

(iii)

a = 9, c = 15

Answer

b = c2−a2\sqrt{c² - a²} = 225−81\sqrt{225 - 81} = 144\sqrt{144} = 12 units

(iv)

a = 7, b = 12

Answer

c = 72+122\sqrt{7² + 12²} = 49+144\sqrt{49 + 144} = 193\sqrt{193} ≈\approx 13.89 units

(v)

a = 1.5, b = 3.5

Answer

c = 1.52+3.52\sqrt{1.5² + 3.5²} = 2.25+12.25\sqrt{2.25 + 12.25} = 14\sqrt{14}.5 ≈\approx 3.81 units

Explanation

(i) Applying Baudhāyana's Theorem: c² = a² + b² = 25 + 49 = 74, so c = 74\sqrt{74}. (ii) Applying Baudhāyana's Theorem: c² = 64 + 144 = 208, so c = 208\sqrt{208} = \sqrt{$$16 \times 13$$} = 413\sqrt{13}. (iii) Rearranging Baudhāyana's Theorem: b² = c² - a² = 225 - 81 = 144, so b = 12. (iv) Applying Baudhāyana's Theorem: c² = 49 + 144 = 193, so c = 193\sqrt{193}. (v) Applying Baudhāyana's Theorem: c² = 2.25 + 12.25 = 14.5, so c = 14\sqrt{14}.5.

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