Question

The side of a right -angled triangle containing the right angle are (5x) cm and(3x-1)cm .If its area is 60 cm2 find its perimeter

Answer 1

Area=1/2 x base x height
60 =1/2 x BC x AB
=1/2 x 5X(3X-1)
=1/2 x 5(3X^2-X)
24=3X^2-X
3X^2-X-24=0
3X^2-9X+8X-24=0
3X(X-3) +8(X-3) =0
(3X+8) (X-3) =0
3X+8=0 or X-3=0
X=-8/3 or X=3
but side of a triangle is not negative
Thus, X=3
substituting value of X:
sides of triangle are 15cm and 8cm
(on using Pythagoras theorem):
third side=17cm
Thus Perimeter= 15+8+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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