Question

The sides of a right angled triangle containing the right angle are 5x CM and (3x-1)cm calculate the length of its hypotenuse of the triangle if it's area is 60 CM square

Answer 1

Area of right - angled triangle = 60 cm²

$\bold{Area \: \: of \: \: right - angled \: \: triangle = \frac{1}{2}\times \: height\times \: base}$

Hence,

$\frac{1}{2} \times \: 5x \times ( 3x - 1 ) = 60$

=> 5x( 3x - 1 ) = 120

=> 15x² - 5x - 120 = 0

=> 5( 3x² - x - 24 ) = 0

=> 3x² - x - 24 = 0

=> 3x² - ( 9 - 8 ) x - 24 = 0

=> 3x² - 9x + 8x - 24 = 0

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

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

=> x = 3 [ Taking positive value because side can't be in negative ]

Hence,

$\mathbb{by \: \: pythagoras \: \: theorem}$

Hypotenuse² = ( 5x )² + ( 3x - 1 )²

Hypotenuse² = 25( 3 )² + { 3( 3 ) - 1 ]²

Hypotenuse² = 25(9) + { 9 - 1 }²

Hypotenuse² = 225 + 64

Hypotenuse = √289 cm



Hypotenuse = 17 cm

Answer 2

Consider ABC as a right angled triangle

AB = 5x cm and BC = (3x – 1) cm

We know that

Area of △ABC = ½ × AB × BC

Substituting the values

60 = ½ × 5x (3x – 1)

By further calculation

120 = 5x (3x – 1)

120 = 15x2 – 5x

It can be written as

15x2 – 5x – 120 = 0

Taking out the common terms

5 (3x2 – x – 24) = 0

3x2 – x – 24 = 0

3x2 – 9x + 8x – 24 = 0

Taking out the common terms

3x (x – 3) + 8 (x – 3) = 0

(3x + 8) (x – 3) = 0

Here

3x + 8 = 0 or x – 3 = 0

We can write it as

3x = -8 or x = 3

x = -8/3 or x = 3

x = -8/3 is not possible

So x = 3

AB = 5 × 3 = 15 cm

BC = (3 × 3 – 1) = 9 – 1 = 8 cm

In right angled △ABC

Using Pythagoras theorem

AC2 = AB2 + BC2

Substituting the values

AC2 = 152 + 82

By further calculation

AC2 = 152 + 82

By further calculation

AC2 = 225 + 64 = 289

AC2 = 172

So AC = 17 cm

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