Question

The side of a right angled triangle containing the right angle are 4x cm and (2x-1) cm. If the area of the triangle is 30 cm2. calculate the length of its sides.

Answer 1

$\large\underline{Given:-}$

• Sides of triangle ⇢ 4x , (2x -1)
• Area of triangle ⇢ 30cm²

$\large\underline{To find:-}$

• length of each side.

$\large\underline{Solutions:-}$

• $\: \: \: \: \: Area \: \: of \: \: triangle \: \: \leadsto \: \: \frac{1}{2} \: \times \: base \: \times \: height$

$\: \: \: \: \: \leadsto \: \: {60} \: \: = \: \: \frac{1}{2} \: \times \: {4x} \: \times \: {({2x} \: - \: {1})}$

$\: \: \: \: \: \leadsto \: \: {30} \: \: = \: \: \frac{1}{2} \: \times \: {4x} \: \times \: {({2x} \: - \: {1})}$

$\: \: \: \: \: \leadsto \: \: {60} \: \: = \: \: {4x} \: \times \: {({2x} \: - \: {1})}$

$\: \: \: \: \: \leadsto \: \: \frac{60}{4} \: \: = \: \: {x} \: \times \: {({2x} \: - \: {1})}$

$\: \: \: \: \: \leadsto \: \: \frac{60}{4} \: \: = \: \: {x} \: \times \: {({2x} \: - \: {1})}$

$\: \: \: \: \: \leadsto \: \: {15} \: \: = \: \: {2x}^{2} \: - \: {x}$

$\: \: \: \: \: \leadsto \: \: {2x}^{2} \: - \: {1} \: - \: {15}$

»★ Now, Solving equation for x.

$\: \: \: \: \: \leadsto \: \: {2x}^{2} \: - \: {6x} \: + \: {5x} \: - \: {15}$

$\: \: \: \: \: \leadsto \: \: {2x} \: {({x} \: - \: {3})} \: - \: {5x} \: {({x} \: - \: {15})}$

$\: \: \: \: \: \leadsto \: \: {({2x} \: + \: {5})} \: \: \: {({x} \: - \: {3})}$

$\: \: \: \: \: \leadsto \: \: {x} \: \: = \: \: \frac{-5}{2} \: \: \: {x} \: \: = \: \: {3}$

»★ Length of any side can't be negative.

So, x ⇢ 3

Base ⇢ 4x

⇢ 4 × 3

⇢ 12 cm

Height ⇢ 2x - 1

⇢ 2 × 3 - 1

⇢ 6 + 1

⇢ 5 cm

✰ The triangle is right angle triangle.

So,

⇢ h² = base² + perpendicular²

⇢ h² = (12)² + (15)²

⇢ h² = 144 + 225

⇢ h² = 169

⇢ h = √169

⇢ h = 13 cm

✰ Hence, The Right angle triangle are sides 12cm, 5cm, and 13cm.

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