Question

＠QualityQuestion

The ratio of the measures of three sides of a triangle is 4 ratio 2 ratio 3 and its perimeter is 36 cm find the area of the triangle.​

Answer 1

Answer:

let the three sides of the triangle are 4xcm,2x cm,3xcm

so perimeter=(4x+2x+3x) cm=9xcm

so, 9x=36

=> x=36/9=4

so, sides are

4×4cm=16 cm

2×4cm=8cm

3×4cm=12cm

half peri.eter =(36 /2)cm=18 cm

area=

√{18×(18-16)(18-8)(18-12) sq cm

=√(18×2×10×6) sq cm

=√(2×3×3×2×2×5×2×3) sq cm

=2×2×3√15 sq cm

=12√15 sq cm

Answer 2

Answer:

Here, the concept of Heron's formula and Perimeter of triangle has been used. We see that we're given the perimeter of triangle and its sides in ratio. Then firstly we can find the sides of triangle. Then we can find the semi-perimeter of the triangle and after finding semi-perimeter we can find the area of triangle by using Heron's formula.

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀

Formula used:

$\bullet\quad\boxed{\sf{Perimeter\; of\; triangle=}\bold{Sum\; of\; all\; sides}}$

$\bullet\quad\boxed{\sf{Semi-perimeter\; of\; triangle=}\bold{\dfrac{Perimeter}{2}}}$

$\bullet\quad\boxed{\sf{Area\; of\; triangle=}\bold{\sqrt{s(s-a)(s-b)(s-c)}}}$

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀

Solution:

Given,

» Perimeter of triangle = 36 cm

• Let the sides of triangle be 4x, 2x and 3x respectively.
• Let the semi-perimeter of triangle be s.

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀

~ For the sides of triangle ::

We know that,

$\\\; \; \longrightarrow\quad\sf{Perimeter\; of\; triangle=Sum\; of\; all\; sides}$

By applying values, we get:

$\\\; \; \longrightarrow\quad\sf{36=a+b+c}$

$\\\; \; \longrightarrow\quad\sf{36=4x+2x+3x}$

$\\\; \; \longrightarrow\quad\sf{36=9x}$

$\\\; \; \longrightarrow\quad\sf{x=\dfrac{36}{9}}$

$\\\; \; \longrightarrow\quad\sf{x=4\; cm}$

We've calculated the value of x. Now, equating the value of x in the sides of triangle.

$\\\quad\quad\leadsto\quad\tt{4x=4\;(4)=}\;\bold{16\; cm}$

$\\\quad\quad\leadsto\quad\tt{2x=2\;(4)=}\;\bold{8\; cm}$

$\\\quad\quad\leadsto\quad\tt{3x=3\;(4)=}\;\bold{12\; cm}$

Hence, sides of triangle are 16 cm, 8 cm and 12 cm respectively.

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀

~ For value of semi-perimeter of triangle ::

We know,

$\\\; \; \dashrightarrow\quad\sf{Semi-perimeter=\dfrac{Perimeter}{2}}$

By applying the values, we get:

$\\\; \; \dashrightarrow\quad\sf{s=\dfrac{36}{2}}$

$\\\; \; \dashrightarrow\quad\sf{s=}\:\bold{18\; cm}$

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀

~ For the Area of Triangle ::

We know that,

$\\\; \; :\implies\quad\sf{Area\; of\; triangle=\sqrt{s(s-a)(s-b)(s-c)}}$

By applying the values, we get:

$\\\; \; :\implies\quad\sf{Area\; of\; triangle=\sqrt{18(18-16)(18-8)(18-12)}}$

$\\\; \; :\implies\quad\sf{Area\; of\; triangle=\sqrt{18(2)(10)(6)}}$

$\\\; \; :\implies\quad\sf{Area\; of\; triangle=\sqrt{2160}}$

$\\\; \; :\implies\quad\sf{Area\; of\; triangle=\sqrt{{2}^{4}\times {3}^{3}\times {5}^{1}}}$

$\\\; \; :\implies\quad\sf{Area\; of\; triangle={2}^{2}\times {3}^{1}\sqrt{3\times 5}}$

$\\\; \; :\implies\quad\sf{Area\; of\; triangle=12\sqrt{15}}$

Since, √15 = 3.87

$\\\; \; :\implies\quad\sf{Area\; of\; triangle=12(3.87)}$

$\\\; \; :\implies\quad\underline{\boxed{\sf{Area\; of\; triangle=}\;\purple{\bold{46.44\; {cm}^{2}}}}}$

Therefore, the required answer is:

❝ The Area of Triangle is 46.44 cm². ❞