Question

Sides of a triangle are in the ratio 3:4 : 5 and its area is 54 cm². Find the sides of the triangle.​

Answer 1

Answer:

Length of the sides are 9 cm, 12 cm and 15 cm.

Step-by-step explanation:

Let the sides be 3x, 4x and 5x.

By Pythagoras Theorem,

$(5x) {}^{2} = {(3x)}^{2} + {(4x)}^{2}$

$25 {x}^{2} = 9 {x}^{2} + 16 {x}^{2}$

$25 {x}^{2} = 25 {x}^{2}$

LHS = RHS

Hence, the sides satisfy the Pythagoras Theorem.

Hence, the angle between the sides 3x and 4x is a right angle.

Taking 4x as the base, the height will be 3x.

$area = \frac{1}{2} bh = \frac{1}{2} \times 3x \times 4x$

$area = 6 {x}^{2}$

It is given that the area is 54 cm². So,

$6 {x}^{2} = 54 {cm}^{2}$

${x}^{2} = 9 {cm}^{2}$

$(x) {}^{2} = {(3cm)}^{2}$

x=3cm

Length of the first side = 3x = 9cm

Length of the second side = 4x = 12cm

Length of the third side = 5x = 15cm

Answer 2

Answer:

$\large \bf{ \underbrace{ \red{ ✪ { \green { \: Given : }}}}}$

Ratio of sides of triangle

3 : 4 : 5

&

Area of the given triangle = 54 cm²

$\large \bf{ \underbrace{ \green{ ✪ { \red{To \: find: }}}}}$

Sides of the triangle

$\large \bf{ \underbrace{ \pink{ ✪ { \color{blue}{Formula \: used : }}}}}$

Heron's Formula

$\large{\boxed{ \bf{{Area \: = \sqrt{s(s - a)(s - b)(s - c)} }}}}$

Where,

S is the semi perimeter of the triangle.

$\large{ \boxed{ \bf{s = \frac{a + b + c}{2} }}}$

$\large \bf{ \underbrace{ \color{blue}{ ✪ { \pink{ \: Solⁿ :}}}}}$

Let the sides of the triangle be 3x, 4x and 5x

(∵ The sides are in the ratio 3 : 4 : 5)

Perimeter of the triangle

$\large \bf = 3x + 4x + 5x$

$\large \bf = 12x$

∴ Semi perimeter (s) of the triangle

$\large \bf = \frac{12x}{2}$

$\large \bf = 6x$

Area of the triangle

$\large \bf \sqrt{6x(6x - 3x)(6x - 4x)(6x - 5x)} = 54 \: cm^{2}$

$\large \bf \sqrt{6x \times 3x \times 2x \times x} = 54 \: cm^{2}$

$\large \bf \sqrt{36x {}^{4} }= 54 \: cm^{2}$

$\large \bf \sqrt{(6x {}^{2}) {}^{2} }= 54 \: cm^{2}$

$\large \bf 6x {}^{2}= 54 \: cm^{2}$

$\large \bf x {}^{2}= \frac{54 \: cm {}^{2} }{6}$

$\large \bf x {}^{2}= 9 \: cm {}^{2}$

$\large \bf ∴x = 3 \: cm$

Now,

$\large \bf 3x = 9 \: cm$

$\large \bf 4x = 12\: cm$

$\large \bf 5x = 15\: cm$

Hence, the sides of the given triangle are 9 cm, 12 cm and 15 cm.

$\large \bf {\red{Glad \: to \: help \: you...}}$

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