Question

. The difference between the sides at right angles in a right-angled triangle is 14 cm.

The area of the triangle is 120 cm2

. Calculate the perimeter of the triangle.

Answer 1

The answer i had written in the notebook
here are its pictures
hope it had been helpfull
The perimeter is 60cm

Answer 2

$\underline\mathfrak{Given:-}$

• A right angle triangle is 14cm.
• The area of the triangle is 120cm².

$\underline\mathfrak{To \: \: Find:-}$

• Find the perimeter of the triangle ......?

$\underline\mathfrak{Solutions:-}$

• Let the side containing right angle be x cm and (x - 14) cm.

Then, it's area

$\: \: \: \: \: = \: \: {[\frac{1}{2} \: \times \: x \: \times \: {(x \: - \: {14})}]} \: {cm}^{2}$

Given, Area = 120 cm²

$\: \: \: \: \: \leadsto \: \: \frac{1}{2} \: \times \: x \: \times \: {(x \: - \: {14})} \: \: = \: \: {120}$

$\: \: \: \: \: \leadsto \: \: {x}^{2} \: - \: {14x} \: \: = \: \: {240}$

$\: \: \: \: \: \leadsto \: \: {x}^{2} \: - \: {14x} \: - \: {240} \: \: = \: \: {0}$

$\: \: \: \: \: \leadsto \: \: {x}^{2} \: - \: {24x} \: + \: {10x} \: - \: {240} \: \: = \: \: {0}$

$\: \: \: \: \: \leadsto \: \: {x} \: {({x} \: - \: {24})} \: + \: {10} \: {({x} \: - \: {24})} \: \: = \: \: {0}$

$\: \: \: \: \: \leadsto \: \: {({x} \: + \: {10})} \: \: \: {({x} \: - \: {24})}$

$\: \: \: \: \: \: \leadsto \: \: {x} \: \: = \: \: {-10} \: \: \: and \: \: \: {x} \: \: = \: \: {24}$

$\: \: \: \: \: \: \: \: \: {x} \: \: = \: \: {24} \: \: \: but \: \: \: {x} \: \: \neq \: \: {-10}$

Other side = (x - 14)

= 24 - 14

= 10cm.

By the Pythagoras theorem:

$\: \: \: \: \: \leadsto \: \: {h} \: \: = \: \: \sqrt{{p}^{2} \: + \: {b}^{2}}$

$\: \: \: \: \: \leadsto \: \: {h} \: \: = \: \: \sqrt{{(24)}^{2} \: + \: {(10)}^{2}}$

$\: \: \: \: \: \leadsto \: \: {h} \: \: = \: \: \sqrt{{576} \: + \: {100}}$

$\: \: \: \: \: \leadsto \: \: {h} \: \: = \: \: \sqrt{676}$

$\: \: \: \: \: \leadsto \: \: {h} \: \: = \: \: {26} \: cm$

Now,

• a = 24
• b = 10
• c = 26

Perimeter of triangle = (a × b × c)

= 24 + 10 + 26

= 60 cm.

Hence, the perimeter of triangle is 60cm.