Question

The perimeter of a rectangle is 106 cm. Its length is (4x-4cm) and
breadth is (2x-9), find its length and breadth.​

Answer 1

The length and breadth of the rectangle are 40 cm and 13 cm, respectively.

Step-by-step explanation:

${\underline{\underline{\bf{Given:}}}}$

• Perimeter of a rectangle = 106 cm
• Length of the rectangle = (4x – 4) cm
• Breadth of the rectangle = (2x – 9) cm

${\underline{\underline{\bf{To\:find:}}}}$

• The actual measurements of length and breadth of the rectangle.

${\underline{\underline{\bf{Formula\:to\:be\:used:}}}}$

$\underline{\boxed{\sf{{Perimeter}_{(Rectangle)}=2(l+b)\:units}}}$

$\:\:\:\:\:\:\:\:\sf{\bullet\:l=length}$

$\:\:\:\:\:\:\:\:\sf{\bullet\:b=breadth}$

${\underline{\underline{\bf{Solution:}}}}$

We are said that the length of a rectangle is (4x – 4) cm and it's breadth is (2x – 9). And the perimeter is 106 cm. The measures of length and breadth are in the form of expressions.

We've to find the actual measurements of length and breadth. And we can find it by inserting the given data in the formula of perimeter of rectangle and solving for the variable 'x'. Let's find it!

$\\ \leadsto{\bf{\purple{2(l+b)\: units=Perimeter}}}$

• Inserting the measures-

$\\ \:\:\:\:\:\::\implies{\sf{106 = 2\times [(4x-4)+(2x-9)]}}$

$\:$

• Grouping the like terms-

$\\ \:\:\:\:\:\:\:\:\::\implies{\sf{106=2\times (4x+2x-4-9)}}$

$\:$

• Simplifying the like terms-

$\\ \:\:\:\:\:\:\:\:\:\:\:\:\::\implies{\sf{106=2\times 6x-13}}$

$\:$

• Transposing the like term from R.H.S to L.H.S (division)-

$\\ \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies{\sf{\dfrac{106}{2}=6x-13}}$

$\:$

• Simplifying the L.H.S-

$\\ \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies{\sf{\dfrac{\cancel{106}}{\cancel{2}}=6x-13}}$

$\\ \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies{\sf{53=6x-13}}$

$\:$

• Again transposing the like term from R.H.S to L.H.S (addition)-

$\\ \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies{\sf{53+13=6x}}$

$\:$

• Simplifying the L.H.S-

$\\ \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies{\sf{66=6x}}$

$\:$

• Transposing the like term to L.H.S (division)-

$\\ \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies{\sf{\dfrac{66}{6}=x}}$

$\:$

• Simplifying L.H.S-

$\\ \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies{\sf{\dfrac{\cancel{66}}{\cancel{6}}=x}}$

$\\ \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies{\boxed{\bf{11=x}}}$

$\\ \:\:\:\:\:\:\:\:\:\:\rightarrowtail{\underline{\underline{\sf{\orange{The\:value\:of\:\bf{x}\:\sf{is}\:\bf{11}}}}}}$

$\:$

So, the measure of:

$\:\:\:\:\:\:\:\bf{\star\:Length\:(4x - 4) \rightarrow \sf{4\times 11-4=44-4}=\underline{\underline{\frak{\red{40\:cm}}}}}$

$\:\:\:\:\:\:\:\bf{\star\:Breadth\:{(2x - 9) \rightarrow \sf{2\times 11-9=22-9}=\underline{\underline{\frak{\red{13\:cm}}}}}}$

$\:$

${\underline{\underline{\bf{Verification:}}}}$

To verify, substitute the obtained measures of length and breadth in the formula of perimeter of rectangle.

$\\ \:\:\:\:\:\mapsto{\sf{2(l+b)=Perimeter}}$

$\\ \:\:\:\:\:\mapsto{\sf{2\times (40+13)=106}}$

$\\ \:\:\:\:\:\mapsto{\sf{2\times 53=106}}$

$\\ \:\:\:\:\:\mapsto{\sf{106=106}}$

$\\ \:\:\:\:\:\mapsto{\sf{L.H.S=R.H.S}}$

$\\ \:\:\:\:\:\mapsto{\sf{Hence,\:our\:answer\:is\: correct!}}$

________________________________________

Answer 2

AnswEr-:

• $\underline{\boxed{\star{\sf{\purple{Length \:of \:Rectangle \:-:\:40cm}}}}}$
• $\underline{\boxed{\star{\sf{\purple{Breadth \:of \:Rectangle \:-:\:13cm}}}}}$

Explanation-:

$\underline{\mathrm{Given-: }}$

• The perimeter of a rectangle is 106 cm.

• The length of Rectangle is (4x-4)cm.

• The Breadth of Rectangle is (2x - 9) cm

$\underline{\mathrm{To\: Find -: }}$

• The actual length and breadth of Rectangle.

$\dag{\underline{\mathrm{Solution\:of\:Question\:-: }}}$

• $\underbrace {\sf{Understanding \:the\:Concept \:-:}}$

• We have to find the actual length and breadth of Rectangle ,

• For this we have to put the given terms in formula of Perimeter of Rectangle.

$\underline{\boxed{\star{\sf{\red{Perimeter \:of \:Rectangle \:-:\:2( Length+ Breadth)\:\:units}}}}}$

• $\underline{\mathrm{Here-: }}$

• The perimeter of a rectangle is 106 cm.
• The length of Rectangle is (4x-4)cm.

• The Breadth of Rectangle is (2x - 9) cm

$\underline{\mathrm{Now,\: Putting \:known\:values\:-: }}$

• $\longrightarrow {\sf{ \:2( (4 x -4+2x-9))=106cm }}$

• $\longrightarrow {\sf{ \:2( 4 x -4+2x-9)=106cm }}$

• $\longrightarrow {\sf{ \:2( 4 x+2x -4 -9)=106cm }}$

• $\longrightarrow {\sf{ \:2( 6x - 13)=106cm }}$

• $\longrightarrow {\sf{ \:( 6x - 13)=\dfrac{\cancel{106}}{\cancel{2}} }}$

• $\longrightarrow {\sf{ \: 6x - 13=53 }}$

• $\longrightarrow {\sf{ \:6x= 53 +13=66}}$

• $\longrightarrow {\sf{ \:6x =66 }}$

• $\longrightarrow {\sf{ \:x=\dfrac{\cancel {66}}{\cancel{6}}}}$

• $\longrightarrow {\sf{ \:x=11 }}$

$\underline{\mathrm{Therefore -: }}$

• $\boxed {\sf{ \:x=11 }}$

$\underline{\mathrm{Now,\: Putting \:x =11\:=\:-: }}$

• The length of Rectangle is $\sf{(4x-4) = 4 \times 11 - 4 = 44 - 4 = 40 cm }$
• The Breadth of Rectangle is $\sf{(2x-9) = 2 \times 11 - 9 = 22 - 9 = 13 cm }$

$\underline{\mathrm{Hence -: }}$

• $\underline{\boxed{\star{\sf{\purple{Length \:of \:Rectangle \:-:\:40cm}}}}}$

• $\underline{\boxed{\star{\sf{\purple{Breadth \:of \:Rectangle \:-:\:13cm}}}}}$

____________________________________

$\huge{\mathrm{Verification ♡ -: }}$

• $\underline{\boxed{\star{\sf{\red{Perimeter \:of \:Rectangle \:-:\:2( Length+ Breadth)\:\:units}}}}}$

$\underline{\mathrm{Here-: }}$

• The perimeter of a rectangle is 106 cm.

• The length of Rectangle is 44cm.

• The Breadth of Rectangle is 13 cm

$\underline{\mathrm{Now,\: Putting \:known\:values\:in\:Formula\:-: }}$

• $\longrightarrow {\sf{ \:2( 44 + 13)=106cm }}$

• $\longrightarrow {\sf{ \:2( 53)=106cm }}$

• $\longrightarrow {\sf{ \:106cm=106cm }}$

$\underline{\mathrm{Therefore -: }}$

• $\boxed {\sf{ \:LHS= RHS }}$

• $\boxed {\sf{ \:Hence \:, Verified}}$

________________________________♡___________________________