Question

a rectangular sheet has an area of 100 square metre and a perimeter of 50 m. find it's diagonals? Explain​

Answer 1

$\large \dag$ Question :-

A rectangular sheet has an area of 100 m² and a perimeter of 50 m. Find it's diagonals.

$\large \dag$ Answer :-

$\red\dashrightarrow\underline{\underline{\sf  \green{The   \: Original \:  Fraction  \: is \:  \sqrt{17}\:m }} }$

$\large \dag$ Step by step Explanation :-

Let us Assume that,

• Lenght of Sheet = $\large \rm l$ m

• Breadth of Sheet = $\large \rm b$ m

❒ We Know that Area of rectangle is :

$\bf \red\bigstar \: \: \orange{ \underbrace{ \underline{  \blue{Area =l \times b }}}}$

So in the given question :

$:\longmapsto \rm Area= l \times b$

$:\longmapsto \rm \large\green{ \underline{ \underline{ \rm l \times b = 100}}} \: - - - (1)$

❒ Also We Know that Perimeter of rectangle is :

$\bf \red\bigstar \: \: \orange{ \underbrace{ \underline{  \blue{Perimeter =2(l + b) }}}}$

So in the given question :

$:\longmapsto \rm Perimeter = 2(l + b)$

$:\longmapsto \rm 2(l + b) = 50$

$:\longmapsto \rm l + b = \cancel\frac{50}{2}$

$:\longmapsto \rm l + b = 25$

$:\longmapsto \rm \large\green{ \underline{ \underline{ \rm l = 25 - b}}}\: - - - (2)$

⏩ Substituting (2) in (1)

$:\longmapsto \rm (25 - b) \times {b} = 100$

$:\longmapsto \rm 25b - {b}^{2} = 100$

$:\longmapsto \rm {b}^{2} - 25b + 100 = 0$

$:\longmapsto \rm b - 5b - 20b + 100 = 0$

$:\longmapsto \rm b(b - 5) - 20(b - 5) = 0$

$:\longmapsto \rm (b - 5)(b - 20) = 0$

$\purple{ \large :\longmapsto  \underline {\boxed{{\bf b = 5 \: \: o r \: \: b = 20} }}}$

⏩ Putting Value of b in (2) :

$\purple{ \large :\longmapsto  \underline {\boxed{{\bf l = 20 \: \: o r \: \: l = 5} }}}$

⇝ The two sides of rectangle are 20 m and 5 m

Therefore,

$:\longmapsto \rm Diagonal = \sqrt{ {20}^{2} + {5}^{2} }$

$:\longmapsto \rm Diagonal = \sqrt{400 + 25}$

$:\longmapsto \rm Diagonal = \sqrt{425}$

$\green{ :\longmapsto  \underline {\boxed{{\bf Diagonal= 5 \sqrt{17} \: m } }}}$

Answer 2

⇝Given :-

• Area of rectangular sheet = 100 m²
• Perimeter of rectangular sheet = 50 m

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⇝To Find :-

• Find its diagonals .

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⇝Formula Used :-

❒ Area of rectangle :

${\large{\red{\bigstar \: \: \: \: \: \: {\orange{\underbrace{\underline{\green{\bf{Area = length \times Breadth }}}}}}}}}$

${\large{\red{\bigstar \: \: \: \: \: \: {\orange{\underbrace{\underline{\green{\bf{Perimeter = 2(length + Breadth ) }}}}}}}}}$

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⇝Solution :-

❒ Let :

• Length = l
• Breadth = b

❒ According to Question :

${\large{:{\longmapsto{\bf{Area = l \times b}}}}}$

${\large{:{\longmapsto{\bf{100 = l \times b}}}}}$

${\large{\orange{\dashrightarrow{\blue{\underline{\bf{l \times b = 100}}}}}}} - - - (1)$

❒ Now :

${\large{:{\longmapsto{\bf{Perimeter = 2(L + B)}}}}}$

${\large{:{\longmapsto{\bf{50= 2(L + B)}}}}}$

${\large{:{\longmapsto{\bf{L + B = {\cancel\frac{50}{2} }}}}}}$

${\large{\orange{\dashrightarrow{\blue{\underline{\bf{l + b = 25}}}}}}} - - - (2)$

❒ Substituting the values :

${\large{:{\longmapsto{\bf{(25 - b) \times b = 100}}}}}$

${\large{:{\longmapsto{\bf{25 b - {b}^{2} = 100}}}}}$

${\large{:{\longmapsto{\bf{ {b}^{2} - 25 + 100= 0}}}}}$

${\large{:{\longmapsto{\bf{b - 5b -20b + 100 = 0}}}}}$

${\large{:{\longmapsto{\bf{b (b - 5) - 20(b - 5) = 0}}}}}$

${\large{:{\longmapsto{\bf{(b - 5)(b - 20)= 0}}}}}$

${\large{\red{:{\twoheadrightarrow{\purple{\underline{\overline{\boxed{\bf{b = 5 , b = 20}}}}}}}}}}$

❒ Diagonal :

${\large{:{\longmapsto{\bf{ \sqrt{ {20}^{2} + {5}^{2} } }}}}}$

${\large{:{\longmapsto{\bf{ \sqrt{20 \times 20 + 5 \times 5} }}}}}$

${\large{:{\longmapsto{\bf{ \sqrt{400 + 25} }}}}}$

${\large{:{\longmapsto{\bf{ \sqrt{425} }}}}}$

${\large{\red{:{\twoheadrightarrow{\purple{\underline{\overline{\boxed{\bf{Diagonal = 5 \sqrt{17} m}}}}}}}}}}$

❒ Hence :

${\huge{\purple{\underline{\red{\underline{\pink{\pmb{\mathfrak{Diagonal = 5 \sqrt{17} m}}}}}}}}}$

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