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The sides of a right-angled triangle containing the right angle are $3 (x + 1) cm$ and $(2 x − 1) cm$ . If the area of the triangle be $30 cm 2$ , then the sides of the triangle are ,

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5cm ,12cm, 13cm

5cm, 12cm, 1cm

2cm, 4cm, 6cm

10cm, 23cm, 12cm

answer is A.

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Detailed Solution

Given,
Area of the triangle is

30 cm 2

and sides are

3 (x + 1)

cm and

(2 x - 1)

cm.
Using the formula,

A r e a = 12 b a s e h e i g h t ⇒ 30 = 12 3 x + 3 2 x − 1 ⇒ 60 = 6 x 2 − 3 x + 6 x − 3 ⇒ 6 x 2 + 3 x − 63 = 0

⇒ 2 x 2 + x − 21 = 0 ⇒ 2 x 2 + 7 x − 6 x − 21 = 0 ⇒ x 2 x + 7 − 3 2 x + 7 = 0 ⇒ x = 3, − 72

x = - \frac{7}{2}

is not possible because side of the triangle is never negative. Thus, the sides are,

3 (x + 1) = 3 (3 + 1) = 12 cm

and

2 x - 1 = 2 (3) - 1 = 5 cm

Let third side of the triangle is

a cm

.
The Pythagorus theorem is given as,

B a s e 2 + P e r p e n d i c u l a r 2 = H y p o t e n u s e 2

Since, given triangle is right-angled triangle then using Pythagoras theorem,

a = 52 + 122 ⇒ a = 13 c m

The sides of the triangle are 5cm,12cm,13cm.
Hence, option 1 is correct.

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