Question

if a diagnol of a rectangle is 53 cm nd one side is 28 cm , find the other side.
A) 25 cm
B) 45 cm
C) 45 sq. cm​
​

Answer 1

Answer:

Diagram :

$\setlength{\unitlength}{1.5cm}\begin{picture}(8,2)\thicklines\put(7.7,3){\sf\large{A}}\put(9.6,1.7){\sf53 cm}\put(7.7,1){\sf\large{B}}\put(9.2,0.7){\sf?}\put(11.1,1){\sf\large{C}}\put(8,1){\line(1,0){3}}\put(8,1){\line(0,2){2}}\put(11,1){\line(0,3){2}}\put(8,3){\line(3,0){3}}\put(11.1,2){\sf28 cm}\put(8,1){\line(3,2){3}}\put(11.1,3){\sf\large{D}}\put(10.8,1){\line(0,2){0.2}}\put(10.8,1.2){\line(2,0){0.2}}\end{picture}$

⠀

Given :

• if a diagnol of a rectangle is 53 cm nd one side is 28 cm

To Find :

• find the other side.

Solution :

$\underline{\bigstar\:\textsf{By Pythagoras Theorem in \triangle BCD :}}$

$:\implies\sf (Hypotenuse)^2=(Base)^2+(Height)^2\\\\\\:\implies\sf (BD)^2=(BC)^2+(CD)^2\\\\\\:\implies\sf (53)^2=(BC)^2+(28)^2\\\\\\:\implies\sf (53)^2-(28)^2=(BC)^2\\\\{\scriptsize\qquad\bf{\dag}\:\:\frak{(a^2-b^2)=(a+b)(a-b)}}\\\\:\implies\sf (53 + 28)(53 - 28) =(BC)^2\\\\\\:\implies\sf 81 \times 25=(BC)^2\\\\\\:\implies\sf \sqrt{81 \times 25} = BC\\\\\\:\implies\sf \sqrt{9 \times 9 \times 5 \times 5} = BC\\\\\\:\implies\sf 9 \times 5 = BC\\\\\\:\implies\underline{\boxed{\sf BC = 45 \:cm}}$

⠀

$\therefore\:\underline{\textsf{Other side of the rectangle is B) \textbf{45 cm}}}.$

Answer 2

(C) 45 cm

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Answer :-

Given :-

• Diagonal of rectangle = 53 cm
• Length of one side is one side = 28 cm

To find :-

• Length of other side = ?

Assume that :-

• There is ABCD rectangle.
• AD is diagonal which is 53 cm.
• BD is of 28 cm which can also be called as Width.
• So, we have to find that Length of rectangle of AB.

We know that :-

• Interior angle of rectangle is equal to 90 degrees.
• So, all interior angle are 90°

In triangle ABD :-

• There is triangle forming ABD in which one angle is 90°, BD which can be also said as Base of triangle is of 28 cm and AD diagonal which can be said as Hypotenuse of triangle is of 53 cm.

So, By Pythagoras Theorem which is applied of triangle ABD :-

${ \boxed{ \color{red}{\mathtt{\implies{Hypotenuse² = Base² + Perpendicular²}}}}}$

[OR]

${ \large{ \boxed{ \color{red}{\mathtt{\implies{(AD)² =( BD)² + (AB)²}}}}}}$

[OR]

${ \large{ \boxed{ \color{red}{\mathtt{\implies{(Diagonal)² = (Breadth)² + (Length)²}}}}}}$

So, length of the other side is :-

${\large{\mathtt{\leadsto{(53)² = (28)² + (Length)²}}}}</p><p>$

${\large{\mathtt{\leadsto{2809 = 784 +(Length)²}}}}$

${\large{\mathtt{\leadsto{2809 - 784 = (Length)²}}}}$

${\large{\mathtt{\leadsto{(Length)² = 2025}}}}$

${\large{\mathtt{\leadsto{Length = \sqrt{ 2025}}}}}$

${ \underline{ \underline{ \color{green}{\large{\mathtt{\leadsto{Length =45 }}}}}}}$

So, length of other side is 45 cm.