Question

The sides of a triangle are 16 cm, 12 cm and 20 cm. Find :
(i) area of the triangle ;
(ii) height of the triangle, corresponding to the largest side ;
(iii) height of the triangle, corresponding to the smallest side.
________________________
kindly don't spam!
best of luck!​

Answer 1

$\large{\boxed{ \sf{ \pink{Question:}}}}$

The sides of a triangle are 16 cm, 12 cm and 20 cm. Find :

(i) area of the triangle ;

(ii) height of the triangle, corresponding to the largest side ;

(iii) height of the triangle, corresponding to the smallest side.

$\large{\boxed{ \sf{ \green{Answer:}}}}$

$\bold{ \orange{part \: 1: }}$

To find area of triangle :

Let sides of triangle be :

a =16cm

b=12cm

c=20cm

$\sf{s = \dfrac{a + b + c}{2} }$

$\implies \: \sf{s = \dfrac{16 + 12 + 20}{2} }$

$\sf{ \implies \: s = \dfrac{48}{2} }$

$\sf{ \implies \: s = 24}$

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

$\sf{Area \: of \: triangle = \sqrt{24(24 - 20)(24- 12)(24 - 16)} }$

$\sf{Area \: of \: triangle = \sqrt{24 \times 4 \times 12 \times 8} }$

$\sf{Area \: of \: triangl e = 96 {cm}^{2} }$

$\bold{ \orange{part \: 2: }}$

To find height corresponding to largest side:

$\sf{Area \: of \: triangle = \dfrac{1}{2} \times base \times height }$

$\sf{96 = \dfrac{1}{2} \times BC \times AD}$

$\sf{96 = \dfrac{1}{2} \times 20\times AD}$

$\sf{AD = \dfrac{96 \times 2}{20} }$

$\sf{AD = 9.6cm}$

$\bold{ \orange{part \: 3: }}$

To find height corresponding to largest side:

$\sf{Area \: of \: triangle = \dfrac{1}{2} \times base \times height }$

$\sf{96 = \dfrac{1}{2} \times AC\times BE}$

$\sf{96 = \dfrac{1}{2} \times 12\times BE}$

$\sf{BE= \dfrac{96 \times 2}{12} }$

$\sf{BE = 16cm}$

Answer 2

Given :

• Area of the triangle with the sides 16 cm, 12 cm and 20 cm is 96 cm².

• The height of the triangle corresponding to the longest side is 9.6 cm.

• The height of the triangle corresponding to the shortest side is 16 cm.

Explanation :

Given :

• Side of the triangle, a = 20 cm
• Side of the triangle, b = 16 cm
• Side of the triangle, c = 12 cm

To find :

• Area of the triangle, A = ?
• Height of the triangle corresponding to the longest side, h = ?
• Height of the triangle corresponding to the shortest side, h = ?

Knowledge required :

• Area of a Scalene triangle :

⠀⠀⠀⠀⠀⠀⠀⠀⠀A = √[s(s - a)(s - b)(s - c)]

Where :

• A = Area of the triangle
• a,b and c = Sides of the triangle
• s = Semi-perimeter of the triangle

Here,

Semi perimeter = s = (a + b + c)/2

• Area of a triangle :

⠀⠀⠀⠀⠀⠀⠀⠀⠀A = ½ × b × h

Where :

• h = Height of the triangle
• b = Base of the triangle

Solution :

Area of the triangle :

First let us find the semi-perimeter of the triangle :

Semi perimeter :

==> s = (a + b + c)

==> s = (20 + 16 + 12)/2

==> s = 48/2

==> s = 24

∴ s = 24 cm

Hence the semi-perimeter of the triangle is 24 cm.

Now using the formula for area of a triangle and substituting the values in it, we get :

==> A = √[s(s - a)(s - b)(s - c)]

==> A = √[24 × (24 - 20) × (24 - 16) × (24 - 12)]

==> A = √(24 × 4 × 8 × 12)

==> A = √9216

==> A = 96

∴ A = 96 cm²

Height of the triangle corresponding to the longest side :

Here,

• Area = 96 cm²
• Base = 20 cm

By using the formula for area of a triangle and substituting the values in it, we get :

==> A = ½ × b = h

==> 96 = ½ × 20 × h

==> 96 = 10h

==> 96/10 = h

==> 9.6 = h

∴ h = 9.6 cm

Height of the triangle corresponding to the shortest side :

Here,

• Area = 96 cm²
• Base = 12 cm

By using the formula for area of a triangle and substituting the values in it, we get :

==> A = ½ × b = h

==> 96 = ½ × 12 × h

==> 96 = 6h

==> 96/6 = h

==> 16 = h

∴ h = 16 cm

Therefore,

• Area of the triangle, A = 96 cm²
• Height of the triangle corresponding to the longest side, h = 9.6 cm
• Height of the triangle corresponding to the shortest side, h = 16 cm²