Question

The lengths of the sides of a triangle are in a ratio of 3 : 4 : 5 and its perimeter is 144 cm. Find the area of the triangle and the height corresponding to the longest side?

Answer 1

hi mate,

answer :The area of triangle is 864 cm² and the height corresponding to the longest side is 28.8 cm.

Step-by-step explanation:

The length of the sides of a triangle are in the ratio 3:4:5. Let the length of sides be 3x,4x,5x.

It is given that the perimeter of the triangle is 144 cm.

Let

the side = 3 x 4 x 5 x

144 = 3x + 4 x + 5 x

x = 144 / 12

x = 12

now we multiply by

3* 12 = 36 Cm = a

4* 12=48cm = b

5* 12=60cm = c = base....

S = a+b+c/2

36 + 48 + 60/2

144/2 = 72

triangle area = √ s (s - a) (s-b) ( s-c)

triangle area

= √ 72 (72-36) (72-48) (72-60)

triangle area = √72 (36) (24) (12)

triangle area = √746496

triangle area=864cm²

triangle area = ½ * base * height

864 = ½* 60 *height

864 = 30 *height

height = 864/30 = 28.8 cm

The height corresponding to the longest side is 28.8 cm.

i hope it helps you..

Answer 2

$\Large{\textbf{\underline{\underline{According\;to\;the\;Question}}}}$

Here we have,

Ratio = 3 : 4 : 5

Perimeter = 144 cm

Assume,

Sides,

a = 3p , b = 4p and c = 5p

Now,

Perimeter of Triangle = a + b + c

144 = 3p + 4p + 5p

12p = 144

p = 144/12

p = 12

Here,

Sides of triangle are :-

a = 3p = 3 × 12 = 36 m

b = 4p = 4 × 12 = 48 m

c = 5p = 5 × 12 = 60 m

Now,

Semi Perimeter of Triangle,

s = (a + b + c)/2

s = (36 + 48 + 60)/2

s = 144/2

s = 72 m

Now,

Using Heron formula :-

A = √s(s - a)(s - b)(s - c)

A = √72(72 - 36)(72 - 48)(72 - 60)

A = √72 × (36) × (24) × (12)

A = √(36 × 2) (36) (12 × 2) × 12

A = √(36 × 36 × 12 × 12) × (2 × 2)

A = 36 × 12 × 2

A = 72 × 12

A = 864 cm²

Now, here,

Longest side = 60 cm

A = ½ x Base x altitude

864 = ½ × 60 × altitude

864 = 30 × altitude

altitude = 864 / 30

altitude = 28.8 cm