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ABD is a right-angled triangle, whose ∠D is the right angle. C is any point on the side BD. If AB = 8 cm, BC = 6 cm and AC = 3 cm, then the length of CD is :

  1. 17121\dfrac{7}{12} cm

  2. 71127\dfrac{1}{12} cm

  3. 122712\dfrac{2}{7} cm

  4. 121712\dfrac{1}{7} cm

Pythagoras Theorem

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Answer

ABD is a right-angled triangle, whose ∠D is the right angle. C is any point on the side BD. If AB = 8 cm, BC = 6 cm and AC = 3 cm, then the length of CD is. Pythagoras Theorem, R.S. Aggarwal Mathematics Solutions ICSE Class 9.

Let CD = x cm

BD = BC + CD = (6 + x) cm

In △ ABD, using Pythagorean theorem,

Hypotenuse2 = Base2 + Height2

⇒ AB2 = BD2 + AD2

⇒ 82 = (6 + x)2 + AD2

⇒ 64 - (6 + x)2 = AD2

⇒ AD2 = 64 - (6 + x)2

⇒ AD2 = 64 - (36 + x2 + 12x)

⇒ AD2 = 64 - 36 - x2 - 12x

⇒ AD2 = 28 - x2 - 12x …..(1)

In △ ADC, using Pythagorean theorem,

⇒ AC2 = CD2 + AD2

⇒ 32 = x2 + AD2

⇒ 9 = x2 + AD2

⇒ AD2 = 9 - x2 …..(2)

From eq.(1) and (2), we have :

⇒ 9 - x2 = 28 - x2 - 12x

⇒ 9 = 28 - 12x

⇒ 12x = 28 - 9

⇒ 12x = 19

⇒ x = 1912\dfrac{19}{12}

⇒ x = 17121\dfrac{7}{12} cm.

Hence, option 1 is the correct option.

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