Question

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

If the area of the triangle be 60 cm2

, calculate the length of the sides of the triangle​

Answer 1

Answer:

Step-by-step explanation:

Area of a right angled triangle

=  

2

1

×b×h

⇒60=  

2

1

×(5x)×(3x−1)

⇒24=x(3x−1)

⇒3x  

2

−x−24=0

⇒3x  

2

−9x+8x−24=0

⇒3x(x−3)+8(x−3)=0

∴x=−  

3

8  

, 3

For x=3, AB=3(3)−1=9−1=8

BC=5×3=15

AB  

2

+BC  

2

=AC  

2

⇒AC  

2

=8  

2

+15  

2

=64+225=289

∴AC=17 units(cm)

∴ perimeter =AB+BC+CA

=8+15+17=40cm.

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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