Question

the sides(in cm) of a right triangle sides containing the right angle are 5x and 5x-7 if the area of right triangle is 60 cm^2 find the sides of the triangle​

Answer 1

Given:-

• The sides (in cm) of a right angled triangle, containing the right angle are 5x and 5x-7.

• The area of right angled triangle is 60cm².

To Find:-

• Sides of the Triangle

Formula Used:-

• ${\boxed{\bf{Area\:of\: Right\: angled\: Triangle=\dfrac{1}{2}\times Base \times Height}}}$

• ${\boxed{\bf{Pythagoras\: theorem:Hypotenuse^2=Height^2+Base^2}}}$

Solution:-

Using Formula,

$\bf :\implies\:Area\:of\: Right\: angled\: Triangle=\dfrac{1}{2}\times Base \times Height$

Let the base be 5x-7 and height be 5x,

Putting values,

$\sf :\implies\:60=\dfrac{1}{2}\times (5x-7) \times 5x$

$\sf :\implies\:60\times2= 25x^2-35x$

$\sf :\implies\: 25x^2-35x-120=0$

$\sf :\implies\: 5x^2-7x-24=0$

Using Middle Term Split,

$\sf :\implies\: 5x^2-(15-8)x-24=0$

$\sf :\implies\: 5x^2-15x+8x-24=0$

$\sf :\implies\: 5x(x-3)+8(x-3)=0$

$\sf :\implies\: (x-3)(5x+8)=0$

$\sf :\implies\: x-3=0\:and\:5x+8=0$

$\sf :\implies\: x=3\:and\:x=\dfrac{-8}{5}$

The value of x can't be negative. So, x = 3

• Base = 5x - 7 = 5×3 - 7 = 15 - 7 = 8cm

• Height = 5x = 5×3 = 15cm

Using Pythagoras theorem,

$\sf :\implies\: Hypotenuse^2=Height^2+Base^2$

$\sf :\implies\: Hypotenuse^2=15^2+8^2$

$\sf :\implies\: Hypotenuse^2=225+64$

$\sf :\implies\: Hypotenuse^2=289$

$\sf :\implies\: Hypotenuse=\sqrt{289}$

$\sf :\implies\: Hypotenuse=17\:cm$

Hence, The sides of the Triangle are 8cm, 15cm and 17cm.

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