Question

A triangle with side lengths 'a' inches, 'b' inches, and 'c' inches has a perimeter of 25 inches. If side a is twice that of side b, and side b is half of side c, what is the length of side b?

Answer 1

Answer :-

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

Here the concept of Perimeter of Triangle has been used. Perimeter of a triangle is the sum of all the sides of triangle. Here we see all the sides are made to depend on others. We can take them as variable quantities and then find their values.

For figure part, refer to Figure section here.

Let's do it !!

_______________________________________________

★ Formula Used :-

$\: \\ \: \large{\boxed{\boxed{\sf{Perimeter \: of \: Triangle_{(\Delta)} \: = \: \bf{Sum \: of \: all \: the \: sides \: \: = \: \: a \: + \: b \: + \: c}}}}}$

$\: \\ \: \large{\boxed{\boxed{\sf{2b \; + \; 2b \; + \; b \; = \; \bf{Perimeter \: of \: Triangle_{(\Delta)}}}}}}$

_______________________________________________

★ Question :-

A triangle with side lengths 'a' inches, 'b' inches, and 'c' inches has a perimeter of 25 inches. If side a is twice that of side b, and side b is half of side c, what is the length of side b ?

_______________________________________________

★ Solution :-

Given,

» Perimeter of the triangle = 25 cm

» Sides of triangle = a, b and c

» Side a is twice than side b

» Side b is half than side c. So, side c is twice the side c.

Clearly, we see here that :-

• a = 2b

• c = 2b

Using this in the perimeter of the triangle, we get,

$\: \\ \qquad \: \large{\sf{\Longrightarrow \: \: \: Perimeter \: of \: Triangle \: = \: \bf{Sum \: of \: all \: the \: sides \: \: = \: \: a \: + \: b \: + \: c}}}$

$\: \\ \qquad \: \large{\sf{\Longrightarrow \: \: \: 2b \; + \; 2b \; + \; b \; = \; \bf{25 \: \: Inches}}}$

$\: \\ \qquad \: \large{\sf{\Longrightarrow \: \: \: 5b \; = \; \bf{25 \: \: inches}}}$

$\: \\ \qquad \: \large{\sf{\Longrightarrow \: \: \: b \; = \; \bf{\dfrac{\cancel{25 \: \: inches}}{\cancel{5}} \: \: = \: \: \underline{\underline{5 \: \: Inches}}}}}$

$\: \\ \: \large{\boxed{\underbrace{\tt{Length \: \: of \: \: Side \: \: 'b' \: \: = \: \: \bf{5 \: \: Inches}}}}}$

Now let us apply the value of 'b', to find the ither sides.

~ For side 'a' :-

➣ a = 2b = 2 × 5 Inches = 10 inches

$\: \\ \: \large{\boxed{\underbrace{\tt{Length \: \: of \: \: Side \: \: 'a' \: \: = \: \: \bf{10 \: \: Inches}}}}}$

~ For side 'c' :-

➣ c = 2b = 2 × 5 Inches = 10 Inches

$\: \\ \: \large{\boxed{\underbrace{\tt{Length \: \: of \: \: side \: \: 'c' \: \: = \: \: \bf{10 \: \: Inches}}}}}$

$\: \\ \: \large{\underline{\underline{\sf{\leadsto \: \: \: Thus, \: length \: of \: side \: \bf{'b'} \: \: is \: \: \boxed{\boxed{\bf{5 \: \: Inches}}}}}}}$

_______________________________________________

$\: \\ \: \qquad \: \large{\underline{\rm{\mapsto \: \: Confused? \: Don't \: worry \: let's \: verify \: it \: :-}}}$

For verification we need to simply apply the values we got into the equations we formed. Then,

$\: \\ \: \large{\sf{\longrightarrow \: \: \: Perimeter \: of \: Triangle \: = \: Sum \: of \: all \: the \: sides \: \: = \: \: a \: + \: b \: + \: c}}$

$\:\: \\ \large{\sf{\longrightarrow \: \: \: 10 \: inch \; + \; 10 \: inch\; + \; 5 \: inch \; = \; 25 \: \: Inches}}$

$\:\: \\ \large{\bf{\longrightarrow \: \: \: 25 \: \: Inches \; = \; 25 \: \: Inches}}$

Clearly, LHS = RHS. So our answer is correct.

Hence, Verified.

_______________________________________________

$\: \: \\ \large{\underbrace{\underline{\sf{\longmapsto \: \: \: Let's \: \: understand \: \: more \: \: :-}}}}$

Evaluation From the Figure :-

Here we see that two sides of the triangle are equal. This means that the given triangle is an Isosceles Triangle.

Since the side a and c are equal and the third side b is of shorter length. Please refer to attachment for the figure of this triangle and better understanding.

From figure :-

• AC = side c = 10 cm

• AB = side a = 10 cm

• BC = side b = 5 cm

=> FG shows length of AC

=> DE shows length of AB

=> HI shows length of BC.

_______________________________________________

$\: \: \\ \large{\underline{\underline{\rm{\leadsto \; \; \; Let's \; understand \; more \; :-}}}}$

• Equilateral Triangle = is the triangle formed where all the three sides are equal and all the angles are 60°.

• Scalene Triangle = is the triangle formed whose sum of all angles is equal to 180° but all the sides are of different measurement.

• Angle Sum Property = This property states that sum of all the angles of triangle is equal to 180°.

• Right Angled Triangle = This is triangle whose one angle is equal to 90° and other two angles are acute angles.

• Obtuse Angle = It is a triangle whose one angle is greater than 90° and other two angles are acute.

Answer 2

Given,

A triangle with side lengths 'a' inches, 'b' inches, and 'c' inches has a perimeter of 25 inches. If side a is twice that of side b, and side b is half of side c.

To find :

Length of side b.

Solution :

Let the length of side c be y inches.

∴ Length of side b = y/2 inches

∴ Length of side a = y/2 * 2 = y inches.

Now atq,

⇒ y + y/2 + y = 25

⇒ (2y + y + 2y)/2 = 25

⇒ 5y = 50

⇒ y = 50/5

⇒ y = 10 inches

Now sides of triangle :

⇒ Length of side a = y = 10 inches

⇒ Length of side b = y/2 = 10/2 = 5 inches

⇒ Length of side c = y = 10 inches

Therefore,

Length of side b = 5 inches.