Question

the length of the side forming a right angle of a right triangle are 5x cm and (3x-1)cm. If the area of the triangle is 60 cm square. find the hypotenuse.

Answer 1

Lengths of side forming the right angle are 5x and 3x-1.
Area of right triangle = 1/2 × Product of lengths of sides forming the right angle.
Area = 60.
So, 60 = 1/2 × 5x × (3x-1)
120 = 5x ×(3x-1)
120= 15x^2 - 5x
15x^2 - 5x - 120 = 0
Dividing the equation by 5,
3x^2 - x -24=0
3x^2 -9x+8x-24=0
3x (x-3) +8 (x-3)=0
(x-3)(3x+8)=0.
So, x-3=0 Or 3x+8=0.
So, x=3 Or x = -8/3.
But length of side cannot be negative.
So, x =3.
Lengths of sides are 5x=5 ×3=15 and 3x-1=3 (3)-1=9-1=8.

By Pythagoras theorem,
15^2 + 8^2 = Hypotneuse ^2
225+64=Hypotenuse ^2
289 = Hypotneuse ^2
Hypotenuse = 17

Answer 2

Hypotenuse of the triangle is 17 cm.

Step-by-step explanation:

Given :-

The length of the sides forming right angle of a right triangle are 5x cm and (3x-1) cm.

Area of the triangle is 60 cm².

To find :-

Hypotenuse of the triangle.

Solution :-

Let the hypotenuse of the triangle be y cm.

And,

let the perpendicular of the triangle be 5x cm and the base of the triangle be (3x-1) cm.

Formula used :

${\boxed{\sf{Area\:of\: right\: triangle=\dfrac{1}{2}\times\:base\times\: perpendicular}}}$

According to the question ,

$\begin{gathered}\to \sf \: \dfrac{1}{2} \times (3x - 1) \times 5x = 60 \\ \\ \to \sf \: (3x - 1)5x = 120 \\ \\ \to \sf \: 15x^{2} - 5x = 120 \\ \\ \to \sf \: 5(3 {x}^{2} - x) = 120 \\ \\ \to \sf \: 3 {x}^{2} - x = 24 \\ \\ \to \sf \: 3 {x}^{2} - x - 24 = 0 \\ \\ \to \sf \: (x - 3)(3x + 8) = 0\end{gathered}$

Either,

x - 3 = 0

→ x = 3

Or,

3x+8=0

→ x = -8/3 [ Impossible]

Therefore,

Perpendicular = 5×3 = 15 cm

Base = (3×3-1) = 8 cm

Now find the hypotenuse by using Pythagoras Theorem.

Hypotenuse² = Perpendicular ² + Base²

→ y ² = 15² + 8²

→ y ² = 225+64

→ y=√289

→ y = 17

Therefore, the hypotenuse of the triangle is 17 cm.