Question

The area of a triangle whose sides
are 4 cm, 13 cm and 15 cm is
equal to the area of a triangle with
base 15 cm. The altitude of the
new triangle will be​

Answer 1

Given:-

• First Side = 4cm

• Second Side = 13cm

• Third Side = 15cm

To Find:-

• The Altitude ( height ) of the triangle.

Formulae used:-

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

Where,

• s = Half of the Perimeter
• a, b, c, = Given Sides.

Now,

→ Perimeter = a + b + c

→ Perimeter = 4 + 13 + 15 → 32

→ s = $\sf{\dfrac{Perimeter}{2}}$

→ s = $\sf{\dfrac{32}{2}= 16 }$

Therefore,

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

→ $\sf{\sqrt{ ( 16 ) ( 16 - 4 ) ( 16 - 13 ) ( 16 - 15 )}}$

→ $\sf{ \sqrt{ ( 16 ) ( 12 ) ( 3 ) ( 1 )}}$

→ $\sf{\sqrt{ 576}}$

→ $\sf{ 24 }$

Thus, The Area of ∆ is 24cm

Now,

→ Area of ∆ = ½ × Base × Height

→ 24 = ½ × 15 × h

→ 24 × 2 = 15h

→ 48 = 15h

→ h = 48/15

→ h = 3.2cm

Hence, The Height of ∆ is 3.2cm.

Answer 2

Given:-

✧ First Side (a) = 4cm

✧ Second Side (b) = 13cm

✧ Third Side (c) = 15cm

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

To Find:-

✧ Altitude of the triangle

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

Solution:-

$\sf \color{fuchsia} We \: know \: that,$

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

$\sf ➝ \: s = \dfrac{4 + 13 + 15}{2}$

$\sf ➝ \: s = \dfrac{32}{2}$

$\sf ➝ \: s = \dfrac{\cancel{32}^{16}}{\cancel{2}_{1}}$

$\sf ➝ \: s = 16$

$\boxed{\sf s = 16cm}$

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

$\sf \color{fuchsia} Now \: we \: know \: that,$

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

$\sf ➝ \: Area = \sqrt{ ( 16 ) ( 16 - 4 ) ( 16 - 13 ) ( 16 - 15 )}$

$\sf ➝ \: Area = \sqrt{ ( 16 ) ( 12 ) ( 3 ) ( 1 )}$

$\sf ➝ \: Area = \sqrt{ 16 × 12 × 3}$

$\sf ➝ \: Area = \sqrt{576}$

$\sf ➝ \: Area = \sqrt{(24)²}$

$\sf ➝ \: Area = 24$

$\boxed{\sf Area = 24 cm²}$

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

$\sf \color{fuchsia}Now \: we \: know \: that,$

$\sf \color{fuchsia} Area \: of \: triangle = \dfrac{1}{2} × Base × Height$

$\sf ➝ \: 24 = \dfrac{1}{2} × 15 × h$

$\sf ➝ \: 24 = \dfrac{15}{2} × h$

$\sf ➝ \: 24 × \dfrac{2}{15} = h$

$\sf ➝ \: \dfrac{48}{15} = h$

$\sf ➝ \: h = \dfrac{48}{15} = 3.2$

$\boxed{\sf h = 3.2cm}$

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

$\sf \color{fuchsia} Therefore, \: height \: of \: triangle \: is \: 3.2cm$