Question

Adjacent sides of a rectangle are in the ratio 5 : 12, if the perimeter of the rectangle is 34 cm, find the length of the diagonal.​​

Answer 1

Answer:

Given: The adjacent sides of a rectangle are in the ratio of 5 : 12. & The perimeter of the rectangle is 34 cm.

Need to find: The Length and Diagonal?

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❍ Let's say, that the Length and Breadth of the given rectangle be 5x and 12x respectively.

⠀⠀⠀

$\underline{\bf{\dag} \:\mathfrak{As\;we\;know\: that\: :}}$⠀⠀⠀⠀

⠀

$\star\;\underline{\boxed{\pmb{\sf{Perimeter_{\:(rectangle)} = 2(Length + Breadth)}}}}$

$\sf{We \;have}\begin{cases}\sf{\quad Length = \bf{5x}}\\\sf{\quad Breadth = \bf{12x}}\\\sf{\quad Perimeter =\bf{34\; cm}}\end{cases}$

$:\implies\sf 34 = 2\Big(5x + 12x\Big)$

$:\implies\sf 34 = 2 \times 17x$

$:\implies\sf 34 = 34x$

$:\implies\sf x = \cancel\dfrac{34}{34}$

$:\implies\underline{\boxed{\pmb{\frak{x = 1}}}}\;\bigstar$

Therefore,

• Length of the rectangle, 5x = 5(1) = 5 cm.
• Breadth of the rectangle, 12x = 12(1) = 12 cm.

⠀

$\therefore{\underline{\textsf{Hence, the Length and Breadth of rectangle are \textbf{5, 12 cm} respectively.}}}$

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⠀

✇ Now, we've to find out the Diagonal of the rectangle. & To Calculate the Diagonal of rectangle formula is Given by —

⠀

$\star\:\underline{\boxed{\pmb{\sf{Diagonal_{\:(rectangle)} = \sqrt{\Big(Length\Big)^2 + \Big( Breadth\Big)^2}}}}}$

» Substituting the Values in the formula —

⠀

$:\implies\sf Diagonal_{\:(rectangle)} = \sqrt{(5)^2 + (12)^2}$

$:\implies\sf Diagonal_{\:(rectangle)} = \sqrt{25 + 144}$

$:\implies\sf Diagonal_{\:(rectangle)} = \sqrt{169}$

$:\implies{ \pmb{\underline{\boxed{\frak{ \mathsf{D}iagonal_{\:(rectangle)} = 13}}}}}\;\bigstar$

$\therefore{\underline{\textsf{Hence, the Diagonal of the rectangle is \textbf{13 cm}.}}}$

Answer 2

Answer:

Answer:

Given: The adjacent sides of a rectangle are in the ratio of 5 : 12. & The perimeter of the rectangle is 34 cm.

Need to find: The Length and Diagonal?

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀

❍ Let's say, that the Length and Breadth of the given rectangle be 5x and 12x respectively.

⠀⠀⠀

\underline{\bf{\dag} \:\mathfrak{As\;we\;know\: that\: :}}

†Asweknowthat:

⠀⠀⠀⠀

⠀

\begin{gathered}\star\;\underline{\boxed{\pmb{\sf{Perimeter_{\:(rectangle)} = 2(Length + Breadth)}}}}\\\\\end{gathered}

⋆

Perimeter

(rectangle)

=2(Length+Breadth)

Perimeter

(rectangle)

=2(Length+Breadth)

\begin{gathered}\sf{We \;have}\begin{cases}\sf{\quad Length = \bf{5x}}\\\sf{\quad Breadth = \bf{12x}}\\\sf{\quad Perimeter =\bf{34\; cm}}\end{cases}\\\\\end{gathered}

Wehave

⎩

⎪

⎪

⎨

⎪

⎪

⎧

Length=5x

Breadth=12x

Perimeter=34cm

\begin{gathered}:\implies\sf 34 = 2\Big(5x + 12x\Big)\\\\\\\end{gathered}

:⟹34=2(5x+12x)

\begin{gathered}:\implies\sf 34 = 2 \times 17x\\\\\\\end{gathered}

:⟹34=2×17x

\begin{gathered}:\implies\sf 34 = 34x\\\\\\\end{gathered}

:⟹34=34x

\begin{gathered}:\implies\sf x = \cancel\dfrac{34}{34}\\\\\\\end{gathered}

:⟹x=

34

34

\begin{gathered}:\implies\underline{\boxed{\pmb{\frak{x = 1}}}}\;\bigstar\\\\\end{gathered}

:⟹

x=1

x=1

★

Therefore,

Length of the rectangle, 5x = 5(1) = 5 cm.

Breadth of the rectangle, 12x = 12(1) = 12 cm.

⠀

\therefore{\underline{\textsf{Hence, the Length and Breadth of rectangle are \textbf{5, 12 cm} respectively.}}}∴

Hence, the Length and Breadth of rectangle are 5, 12 cm respectively.

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⠀

✇ Now, we've to find out the Diagonal of the rectangle. & To Calculate the Diagonal of rectangle formula is Given by —

⠀

\begin{gathered}\star\:\underline{\boxed{\pmb{\sf{Diagonal_{\:(rectangle)} = \sqrt{\Big(Length\Big)^2 + \Big( Breadth\Big)^2}}}}}\\\\\end{gathered}

⋆

Diagonal

(rectangle)

=

(Length)

2

+(Breadth)

2

Diagonal

(rectangle)

=

(Length)

2

+(Breadth)

2

» Substituting the Values in the formula —

⠀

\begin{gathered}:\implies\sf Diagonal_{\:(rectangle)} = \sqrt{(5)^2 + (12)^2} \\\\\\\end{gathered}

:⟹Diagonal

(rectangle)

=

(5)

2

+(12)

2

\begin{gathered}:\implies\sf Diagonal_{\:(rectangle)} = \sqrt{25 + 144}\\\\\\\end{gathered}

:⟹Diagonal

(rectangle)

=

25+144

\begin{gathered}:\implies\sf Diagonal_{\:(rectangle)} = \sqrt{169}\\\\\\\end{gathered}

:⟹Diagonal

(rectangle)

=

169

\begin{gathered}:\implies{ \pmb{\underline{\boxed{\frak{ \mathsf{D}iagonal_{\:(rectangle)} = 13}}}}}\;\bigstar\\\\\end{gathered}

:⟹

Diagonal

(rectangle)

=13

Diagonal

(rectangle)

=13

★

\therefore{\underline{\textsf{Hence, the Diagonal of the rectangle is \textbf{13 cm}.}}}∴

Hence, the Diagonal of the rectangle is 13 cm.