Question

Question -
The sides of a triangle are 25cm, 39cm and 56cm respectively.
Find the Given terms :
( I ) Area of the Triangle.
( II ) The altitude to the longest side of the Triangle.​

Answer 1

AnswEr :

$\bold{Given \:Sides \:of \: \triangle} \begin{cases}\sf{a=25 \:cm} \\ \sf{b=39 \:cm}\\ \sf{c=56 \:cm}\end{cases}$

We will use Heron's Formula to find the Area of Triangle (let's say ∆ABC).

⋆ Refrence of Image is in the Diagram :

$\setlength{\unitlength}{1cm}\begin{picture}(6,8)\linethickness{0.075mm}\put(1, .5){\line(2, 1){3}}\put(4, 2){\line(-2, 1){2}}\put(2, 3){\line(-2, -5){1}}\put(.7, .3){ A }\put(4.05, 1.9){ B }\put(1.7, 2.95){ C }\put(3.2, 2.5){ 25 cm }\put(0.6,1.7){ 39 cm }\put(2.7, 1.05){ 56 cm }\end{picture}$

( I ) Area of the Triangle.

• First we will find the Semi Perimeter :

$\longrightarrow \tt Semi \:Perimeter = \dfrac{Sum \:of \:Sides}{2} \\ \\\longrightarrow \tt s = \dfrac{a + b + c}{2} \\ \\\longrightarrow \tt s = \dfrac{25 + 39 + 56}{2}\\ \\\longrightarrow \tt s = \cancel\dfrac{120}{2} \\ \\\longrightarrow \blue{\tt s = 60}$

$\rule{300}{1}$

• Calculation of Area of Triangle :

$\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{s(s - a)(s - b)(s - c)} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{60(60 - 25)(60 - 39)(60- 56)} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{60 \times35 \times21\times4}\\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{(4 \times3 \times5) \times(7 \times5) \times(7 \times3) \times4} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{(4 \times 4)(5 \times 5)(3 \times 3)(7 \times7)} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= 4 \times5 \times 3 \times 7 \\ \\\longrightarrow \boxed{\orange{\tt Area_{\tiny \triangle ABC}= 420 \:{cm}^{2}}}$

⠀

∴ Area of Triangle will be 420 cm².

$\rule{300}{2}$

( II ) The altitude to the longest side of the Triangle.

• Longest Side will be Base i.e. 56 cm

$\implies \tt Area_{\tiny \triangle ABC} = \dfrac{1}{2} \times Base \times Height \\ \\\implies \tt 420 = \dfrac{1}{ \cancel2} \times \cancel{56} \times Height \\ \\\implies \tt 420 =28 \times Height \\ \\\implies \tt \cancel\dfrac{420}{28} = Height \\ \\\implies \boxed{ \orange{ \tt Height = 15cm}}$

⠀

∴ Altitude to the longest side is 15 cm.

#answerwithquality #BAL

Answer 2

Answer:

STEP 1: Find the Area:

Area = √(p)(p-a)(p-b)(p-c)

p = (25 + 39 + 56) ÷ 2 = 60

Area = √(60)(60-25)(60-39)(60-56)

Area = √176400

Area = 420 cm²

STEP 2: Find the altitude:

Area = 1/2 x base x height

420 = 1/2 x 56 x height

420 = 28 x height

height = 420 ÷ 28

height = 15 cm