Question

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

area of the triangle is 60 cm2

, find its all sides.

Answer 1

area of right angled triangle=½(base) (height)
60cm²=½(5x)(3x-1)
60=½[15x²-5x]
60×2=15x²-5x
15x²-5x-120=0
3x²-x-24=0
3x²-9x+8x-24=0
3x(x-3)+8(x-3)=0
(3x+8)(x-3)=0
x=-8/3{which is not possible as length of side can not be negative}
so, x=3
therefore
sides will be
5x=5(3)=15cm
and (3x-1)={(3×3)-1}=8cm
therefore the third side will be =√[(15)²+(8)²]
=√[225+64]=√289=17cm

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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