Question

The sides of a right triangle containing the right angle are (5x) cm and (3x - 1) cm. If the
area of the triangle be 60 cm", calculate the length of the sides of the triangle.​

Answer 1

Answer:

Given that sides of a right-angled triangle are 5x and (3x - 1)cm.

Given that Area of the triangle = 60cm^2.

We know that Area of the triangle = 1/2 * b * h

                              60 = 1/2 * 5x * (3x - 1)

                             5x(3x - 1) = 60 * 2

                             5x(3x - 1) = 120

                             x(3x - 1) = 120/5

                              3x^2 - x = 24

                              3x^2 - x - 24 = 0

                              3x^2 + 8x - 9x - 24 = 0

                              x(3x + 8) - 3(3x + 8)

                              (x - 3)(3x + 8)

                              x = 3 (or) x = -3/8.

x value should not be -ve.Therefore the value of x = 3.

Therefore the sides of a right-angled triangle = 

5x = 5 * 3 = 15cm

(3x - 1) = (3 * 3 - 1)

           = 9 - 1

           = 8cm

 

By Pythagoras theorem, we know that 

h^2 = 15^2 + 8^2

       = 225 + 64

       = 289

h = 

   = 17.

Therefore the hypotenuse = 17cm.

Therefore the sides of the triangle are 8cm,15cm, and 17cm.

Hope this helps!

Answer 2

$\Large{\textbf{\underline{\underline{According\:to\:the\:Question}}}}$

As we know that :-

$\tt{\rightarrow Area\;of\;Triangle=\dfrac{1}{2}\times base\times height}$

We can also write it as without hypotenuse

$\tt{\rightarrow\dfrac{1}{2}\times Product\;of\;Sides}$

Also,

Sides of the triangle

= 5x cm

Also,

(3x - 1) cm

According to Information,

Area of triangle = 60 cm²

$\tt{\rightarrow\dfrac{1}{2}\times 5x\times(3x-1)=60cm^2}$

5x(3x - 1) = 120

x(3x - 1) = 24

3x² - x = 24

3x² - x - 24 = 0

3x² - (9 - 8)x - 24 = 0

3x² - 9x + 8x² - 24 = 0

3x(x - 3) + 8(x - 3) = 0

(x - 3)(3x + 8) = 0

Now,

x - 3 = 0

x = 3

Also,

3x + 8 = 0

3x = -8

$\tt{\rightarrow x=\dfrac{-8}{3}}$

${\boxed{\sf\:{Negative\;value\;is\;not\;applicable}}}$

Therefore,

Length of sides :-

5x = 5 × 3 = 15 cm

3x - 1 = 3 × 3 - 1 = 8 cm