Question

The perimeter of a triangle is 48 feet. If the second side is 3 more than twice the first side and the third side is 5 more than the first side, then what are the lengths of each side?

Answer 1

Answer :-

$\: \: \boxed{\boxed{\rm{\mapsto \: \: \: Firstly \: let's \: understand \: the \: concept \: used}}}$

Here the concept of Perimeter of Triangle and Linear Equations has been used. We see here that the rest of the sides of Triangle has been made to depend on one side. And perimeter of the triangle is given. So, if we form an equation using this, we can easily solve it.

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★ Formula Used :-

$\: \large{\boxed{\boxed{\sf{Perimeter \: of \: the \: triangle_{(\Delta)} \: = \: \bf{Sum \: of \: all \: the \: sides}}}}}$

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★ Question :-

The perimeter of a triangle is 48 feet. If the second side is 3 more than twice the first side and the third side is 5 more than the first side, then what are the lengths of each side?

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★ Solution :-

Given,

» Perimeter of the triangle = 48 feet

» Second side = 3 + 2(The first side)

» Third side = 5 + The first side

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• Let the first side be 'a' feet

• Let the second side be 'b' feet

• Let the third side be 'c' feet

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Then, according to the question :-

~ Case I :-

➣ b = 3 + 2a ... (i)

~ Case II :-

➣ c = 5 + a .. (ii)

Applying the values of these equations into the perimeter, we get,

➣ a + 3 + 2a + 5 + a = 48

➣ 4a = 48 - 3 - 5

➣ 4a = 40

$\: \\ \: \large{\bf{\Longrightarrow \: \: \: a \: = \: \: \dfrac{40}{4} \: \: = \underline{10}}}$

➣ a = 10

$\: \\ \large{\boxed{\boxed{\tt{a \: \: = \: \bf{10 \: \: feet}}}}}$

Now let us apply the value of a in the equations.

Then, length of all the sides is :-

From equation (i) ;

• b = 3 + 2a = 3 + 2(10) = 3 + 20 = 23

$\: \\ \large{\boxed{\boxed{\tt{b \: \: = \: \bf{23 \: \: feet}}}}}$

From equation (ii) ;

• c = 5 + a = 5 + 10 = 15

$\: \\ \large{\boxed{\boxed{\tt{c \: \: = \: \bf{15 \: \: feet}}}}}$

$\: \\ \qquad \large{\underline{\underline{\rm{\mapsto \: \: \: The \: length \: of \: sides \: of \: the \: given \: triangle_{(\Delta)} \: are \: \boxed{\bf{10 \: \: cm, \: \: 15 \: \: cm \: \: and \: \: 23 \: \: cm}}}}}}$

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$\: \qquad \large{\underbrace{\sf{\leadsto \: \: \: Let's \: know \: more \: :-}}}$

• Area of Square = (Side)²

• Area of Rectangle = Length × Breadth

• Area of Triangle = ½ × Base × Height

• Area of Parallelogram = Base × Height

• Perimeter of Rectangle = 2(Length + Breadth)

• Perimeter of Square = 4 × Side

• Area of Circle = πr²

• Perimeter of Circle = 2πr

Answer 2

Given :

The perimeter of a triangle is 48 feet. If the second side is 3 more than twice the first side and the third side is 5 more than the first side.

To find :

What are the lengths of each side?

Solution :

Let the first side of Δ be 'a' feet.

∴ Second side = (2a + 3) feet

∴ Third side = (a + 5) feet

Now as we know,

⇒  Perimeter of Δ = Sum of 3 sides of Δ

⇒  48 = a + 2a + 3 + a + 5

⇒  48 = 4a + 8

⇒  48 - 8 = 4a

⇒  40 = 4a

⇒  a = 40/4

⇒  a = 10 feet

∴ Length of 1st side = a = 10 feet

∴ Length of 2nd side = 2a + 3 = 2(10) + 3 = 20 + 3 = 23 feet

∴ Length of 3rd side = a + 5 = 10 + 5 = 15 feet

Therefore,

Length of sides of Δ are 10, 23 and 15 feet respectively.