Question

Find the area of the triangle with heron method -:

1. 21cm , 17 cm , 10 cm

2. 42 cm , 39 cm , 45 cm


No spams ​

Answer 1

Answers:

Question 1:

21 cm, 17 cm and 10 cm

a = 21 cm

b = 17 cm

c = 10 cm

S = $\dfrac{a+b+c}{2}$

= $\dfrac{21+17+10}{2}=\dfrac{48}{2}=24cm$

Area =$\sqrt{S(S-a)(S-b)(S-c)}$

=$\sqrt{24(24-21)(24-17)(24-10)}$

=$\sqrt{24\times3\times7\times14}$

=$\sqrt{3\times8\times3\times7\times2\times7}$

= 3 x 7 x 2 x 2

= 84cm²

∴ The area of the triangle is 84cm².

Question 2:

42 cm, 39 cm and 45 cm

a = 42 cm

b =39 cm

c = 45 cm

S = $\dfrac{a+b+c}{2}$

= $\dfrac{42+39+45}{2}=\dfrac{126}{2}=63cm$

Area = $\sqrt{S(S-a)(S-b)(S-c)$

= $\sqrt{63(63-42)(63-39)(63-45)$

=$\sqrt{63\times21\times24\times18$

=$\sqrt{9\times7\times7\times3\times3\times8\times9\times2$

=$3\times3\times3\times7\sqrt{2\times4\times2$

=3 x 3 x 3 x 7 x 2 x 2

= 756cm²

∴ The area of the triangle is 756cm².

Knowledge Bytes:

→ Heron's Formula

The area of a triangle of sides a, b and c is $\sqrt{S(S-a)(S-b)(S-c)$ where semi-perimeter S is the half of the sum of all sides of the triangle. $\dfrac{a+b+c}{2}$.

Answer 2

$\bold{\underline{\underline{Let's \: \: have \: \: a \: \: glance \: \: at \: \: Heron's \: \: Formula}}}$

• Using Heron's Formula we can find the area of any triangle.

✯ Steps for using Heron's Formula :-

(i) Find Semi Perimeter of the triangle.

$\bold{S = \frac {a \: + \: b\: +\: c}{2}}$

(ii) Then use the below ⬇️ formula.

$\bold {Heron's \: \: Formula = \sqrt{s(s - a)(s-b)(s - c)}}$

Here , a ,b and c denotes the different sides of the triangle.

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

${\frak{Question \: 1}}$

a = 21 cm

b = 17 cm

c = 10 cm

$\bold{Semi \: Perimeter = \frac {a \: + \: b\: +\: c}{2}}$

S = (21cm + 17cm + 10cm)/2

S = 48/2 cm

S = 24 cm

$\bold {Heron's \: \: Formula = \sqrt{s(s - a)(s-b)(s - c)}}$

Area = √24 (24 - 21) (24 - 17) (24 - 10)

Area = √24 × 3 × 7 × 14

Area = √2 × 2 × 2 × 3 × 3 × 7 × 7 × 2

Area = √2 × 2 × 2 × 2 × 3 × 3 × 7 × 7

Area = 2 × 2 × 3 × 7 cm²

Area = 84 cm²

∴ The area of the triangle is 84cm².

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

${\frak{Question \: 2}}$

a = 42 cm

b = 39 cm

c = 45 cm

$\bold{Semi \: Perimeter = \frac {a \: + \: b\: +\: c}{2}}$

S = (42cm + 39cm + 42cm)/2

S = 126/2 cm

S = 63 cm

$\bold {Heron's \: \: Formula = \sqrt{s(s - a)(s-b)(s - c)}}$

Area = √63 (63 - 42) (63 - 39) (63 - 45)

Area = √63 × 21 × 24 × 18

Area = √9 × 7 × 7 × 3 × 2 × 2 × 2 × 3 × 9 × 2

Area = √9 × 9 × 7 ×7 × 3 × 3 ×2 × 2 × 2 × 2

Area = 9 × 7 × 3 × 2 × 2 cm²

Area = 756cm²

∴ The area of the triangle is 756cm².

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬