Question

the side of a triangle are 14 cm , 6 cm and 10 cm long.find the area of the triangle​

Answer 1

To Find:-

The area of the triangle

Given:-

Side a = 14cm

Side b = 6cm

Side c = 10cm

Concept:-

Firstly,we have to find the find the semiperimeter of the given ∆ABC with sides a = 14cm,b = 6cm and c = 10cm by adding the all sides and dividing it by 2.Then,We should find the area using the Heron's Formula formula

Formulae Applied:-

$\</u></strong><strong><u>l</u></strong><strong><u>arge\boxed{ \bold{</u></strong><strong><u>S</u></strong><strong><u>emiperimeter \: = \frac{a \: + b \: + c }{2}}}$

$\</u></strong><strong><u>L</u></strong><strong><u>arge\boxed{\bold{</u></strong><strong><u>A</u></strong><strong><u>rea \: = \: \sqrt{s({s \: - \: a})({s \: - \: b})({s \: - \: c)}}}}$

Solution:-

a = 14cm

b = 6cm

c = 10cm

Firstly,we have to find the semiperimeter(S).

$\large\boxed{ \bold{Semiperimeter \: = \frac{a \: + b \: + c }{2}}}$

$\Large{\implies\large\bold{s \: = \frac{a \: + b \: + c }{2}}}$

$\Large{\implies\large\bold{s \: = \frac{14 \: + 6 \: + 10 }{2}}}$

$\Large{\implies\large\bold{s \: = \frac{14 \: + 6 \: + 10 }{2}} \: = \frac{30}{2}}$

$\Large\implies\large\bold{s \: = \frac{30 }{2} \: = \: 15}$

The semiperimeter of the triangle = 15cm

Now,we have to find the area

$\Large\boxed{\bold{Area \: = \: \sqrt{s({s \: - \: a})({s \: - \: b})({s \: - \: c)}}}}$

$\bold{Area \: = \: \sqrt{15({15 \: - \: 14})({15 \: - \: 6})({15 \: - \: 10)}}}$

$\bold{= \: \sqrt{15({1})({9})({5})}}$

$\bold{= \: \sqrt{675}}$

$\bold{= \: \sqrt{675} \: = 25.98cm}$

The area of the triangle = 25.98cm

Additional Information:-

• Semiperimeter is needed to find the area of a triangle using Heron's Formula.

• Heron's Formula is used when we are given the three sides of a triangle and to find its area.

• It is a simple way to find the area of a triangle when the length of all three sides are known or specified.

• Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle firstly.

Answer 2

Given :-

• First side = 14 cm

• Second side = 6 cm

• Third side = 10 cm

To Find :-

The area of the triangle.

Analysis :-

Firstly, find the semi perimeter by substituting the values given in it's respective formula.

Using Heron's formula, find the area accordingly.

Solution :-

We know that,

• s = Semi perimeter
• a = Area

By the formula,

$\underline{\boxed{\sf Semi \ perimeter=\dfrac{a+b+c}{2} }}$

Given that,

• First side (a) = 14 cm

• Second side (b) = 6 cm
• Third side (c) = 10 cm

Substituting their values,

s = 14+6+10/2

s = 30/2

s = 15

Therefore, the semi perimeter of the triangle is 15 cm.

Using Heron's formula,

$\underline{\boxed{\sf Area \ of \ triangle=\sqrt{s(s-a)(s-b)(s-c)} }}$

Given that,

• Semi perimeter (s) = 15 cm

• First side (a) = 14 cm

• Second side (b) = 6 cm

• Third side (c) = 10 cm

Substituting their values,

$\sf =\sqrt{15(15-14)(15-6)(15-10)}$

$\sf =\sqrt{15 \times 1 \times 9 \times 5}$

$\sf =\sqrt{675}$

$\sf = 25.98 \ cm^2$

Therefore, area of the triangle is 25.98 cm².