Question

the perimeter of a triangle is 54 and it's sides are in the ratio 5 ratio 6 ratio 7• find the area of the triangle​

Answer 1

SOLUTION:-

Given:

The perimeter of a triangle is 54cm & its side are in the ratio 5:6:7.

So,

Let the number be x

⚫5x cm

⚫6x cm

⚫7x cm

We know that, perimeter of ∆;

=) side + side + side

=) 5x + 6x + 7x = 54

=) 18x = 54

=) x= 54/18

=) x= 3cm

Now,

1st side,5x= 5×3=15cm

2nd side,6x=6×3=18cm

3rd side,7x= 7×3=21cm

Using Heron's Formula:

⚫A= 15cm

⚫B= 18cm

⚫C= 21cm

$\begin{gathered}s = \frac{a + b + c}{2} \\ \\ = > \frac{15 + 18 + 21}{2} \\ \\ = > \frac{54}{2} \\ \\ = > 27cm\end{gathered} </p><p>$

Area of triangle:

$\begin{gathered}a = \sqrt{s(s - a)(s - b)(s - c)} \\ \\ = > \sqrt{27(27 - 15)(27 - 18)(27 - 21)} \\ \\ = > \sqrt{27(12)(9)(6)} \\ \\ = > \sqrt{3 \times 3 \times 3 \times 2 \times 2 \times 3 \times 3 \times 3 \times 2 \times 3} \\ \\ = > 3 \times 3 \times 3 \times 2 \sqrt{6} \\ \\ = > 54 \sqrt{6} {cm}^{2} \end{gathered}$

Thus,

The area of triangle is 54√6 cm².

Hope it helps ☺️

Answer 2

Step-by-step explanation:

The perimeter of a triangle is 54cm & its side are in the ratio 5:6:7.

So,

Let the number be x

⚫5x cm

⚫6x cm

⚫7x cm

We know that, perimeter of ∆;

=) side + side + side

=) 5x + 6x + 7x = 54

=) 18x = 54

=) x= 54/18

=) x= 3cm

Now,

1st side,5x= 5×3=15cm

2nd side,6x=6×3=18cm

3rd side,7x= 7×3=21cm

Using Heron's Formula:

⚫A= 15cm

⚫B= 18cm

⚫C= 21cm

\begin{gathered}\begin{gathered}s = \frac{a + b + c}{2} \\ \\ = > \frac{15 + 18 + 21}{2} \\ \\ = > \frac{54}{2} \\ \\ = > 27cm\end{gathered} < /p > < p > \end{gathered}

s=

2

a+b+c

=>

2

15+18+21

=>

2

54

=>27cm

</p><p>

Area of triangle:

\begin{gathered}\begin{gathered}a = \sqrt{s(s - a)(s - b)(s - c)} \\ \\ = > \sqrt{27(27 - 15)(27 - 18)(27 - 21)} \\ \\ = > \sqrt{27(12)(9)(6)} \\ \\ = > \sqrt{3 \times 3 \times 3 \times 2 \times 2 \times 3 \times 3 \times 3 \times 2 \times 3} \\ \\ = > 3 \times 3 \times 3 \times 2 \sqrt{6} \\ \\ = > 54 \sqrt{6} {cm}^{2} \end{gathered}\end{gathered}

a=

s(s−a)(s−b)(s−c)

=>

27(27−15)(27−18)(27−21)

=>

27(12)(9)(6)

=>

3×3×3×2×2×3×3×3×2×3

=>3×3×3×2

6

=>54

6

cm

2

Thus,

The area of triangle is 54√6 cm².

Hope it helps ☺️