Question

the length of sides of a triangle are in ratio 3:4:5 and its perimeter is 144m. find the area of the triangle and the height corresponding to the the longest side ?​

Answer 1

AnswEr :

Let us Consider that the sides of the Triangle are 3x, 4x & 5x.

Perimeter of Triangle = 144m. [Given]

______________________

$\longrightarrow\sf 3x + 4x + 5x = 144$

$\longrightarrow\sf 12x = 144$

$\longrightarrow\sf x = \dfrac{144}{12}$

$\longrightarrow\small{\underline{\boxed{\sf{\red{x \:=\: 12}}}}}$

Sides of the Triangle:

$\longrightarrow\sf 3x = 3(12)$

$\longrightarrow\small\boxed{\sf{\pink{36}}}$

$\longrightarrow\sf 4x = 4(12)$

$\longrightarrow\small\boxed{\sf{\pink{48}}}$

$\longrightarrow\sf 5x = 5(12)$

$\longrightarrow\small\boxed{\sf{\pink{60}}}$

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$\dag\:\:\small\bold{\underline{\sf{Now,\: Using\: Heron's\: Formula}}}$

$\longrightarrow\large{\underline{\boxed{\sf{\red{\sqrt{s(s\:-\:a) (s\:-\:b) (s\:-\:c)}}}}}}$

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

$\longrightarrow\sf s = \dfrac{144}{2}$

$\longrightarrow\sf s = 72$

$\sf{Here}\begin{cases} \sf{s \: = \: 72} \\ \sf{a \:=\: 36} \\ \sf{b\:=\: 48} \\ \sf{c\:=\: 60} \end{cases}$

$\longrightarrow\sf \sqrt{72(72 - 36) (72 - 48) (72 - 60)}$

$\longrightarrow\sf \sqrt{72(36) (24) (12)}$

$\longrightarrow\sf\sqrt{746496}$

$\longrightarrow\sf 864$

Area of Triangle is 864 cm².

Now, Height corresponding to the longest side :

$\longrightarrow\sf \dfrac{ 2 \times 864}{60}$

$\longrightarrow\large\boxed{\sf{\red{28.8 \:cm}}}$

Hence,the Height corresponding to the longest side is 28.8 cm.

Answer 2

Answer:

⋆ DIAGRAM :

$\setlength{\unitlength}{1.5cm}\begin{picture}(6,2)\put(7.7,2.9){\large{A}}\put(7.7,1){\large{B}}\put(10.6,1){\large{C}}\put(8,1){\line(1,0){2.5}}\put(8,1){\line(0,2){1.9}}\put(10.5,1){\line(-4,3){2.5}}\put(7.5,2){\sf{\large{4x}}}\put(9,0.7){\sf{\large{3x}}}\put(9.4,1.9){\sf{\large{5x}}}\put(8.2,1){\line(0,1){0.2}}\put(8,1.2){\line(3,0){0.2}}\end{picture}$

Let the Sides be 3x, 4x and 5x. As these are Pythagoras Triplet, then this Triangle must be Right Angled Triangle.

$\underline{\bigstar\:\:\textsf{Perimeter of Triangle ABC :}}$

$:\implies\tt Perimeter=Sum\:of\:Sides\\\\\\:\implies\tt 144 =3x + 4x + 5x\\\\\\:\implies\tt 144 = 12x\\\\\\:\implies\tt \dfrac{144}{12} =x\\\\\\:\implies\tt x = 12$

$\bullet\:\:\textsf{BC = 3x = 3(12) = \textbf{36 m}}\\\bullet\:\:\textsf{AB = 4x = 4(12) = \textbf{48 m}}\\\bullet\:\:\textsf{AC = 5x = 5(12) = \textbf{60 m}}$

$\rule{160}{1}$

$\underline{\bigstar\:\:\textsf{Area of Triangle ABC :}}$

$\dashrightarrow\tt\:\:Area_{{\tiny\triangle ABC}}=\dfrac{1}{2} \times Base \times Height\\\\\\\dashrightarrow\tt\:\:Area_{{\tiny\triangle ABC}} =\dfrac{1}{2} \times 36\:m \times48 \:m\\\\\\\dashrightarrow\tt\:\:Area_{{\tiny\triangle ABC}} =18\:m \times 48 \:m\\\\\\\dashrightarrow\:\:\underline{\boxed{\tt Area_{{\tiny\triangle ABC}} =864 \:m^{2}}}$

$\rule{200}{2}$

$\underline{\bigstar\:\:\textsf{Height Corresponding to Longest Side :}}$

$\dashrightarrow\tt\:\:Area_{{\tiny\triangle ABC}}=\dfrac{1}{2} \times Base \times Height\\\\\\\dashrightarrow\tt\:\:864 \:m^2 = \dfrac{1}{2} \times60 \:m \times Height\\\\\\\dashrightarrow\tt\:\: \dfrac{864 \:m^2 \times 2}{60\:m} = Height\\\\\\\dashrightarrow\tt\:\:\underline{\boxed{\textsf{\textbf{Height = 28.8 m}}}}$