Question

The length and breadth of a rectangle are 24 cm and 7 cm respectively. What must be the length of its diagonal (in cm)?

A) 50 B) 28 C) 25 D) 56​

Answer 1

⋆ DIAGRAM :

$\setlength{\unitlength}{1.5cm}\begin{picture}(8,2)\thicklines\put(7.7,3){\tt\large{A}}\put(7.7,1){ \tt\large{B}}\put(9.5,0.7){\sf{\large{24 cm}}}\put(11.5,1){ \tt\large{C}}\put(8,1){\line(1,0){3.5}}\put(8,1){\line(0,2){2}}\put(11.5,1){\line(0,3){2}}\put(8,3){\line(3,0){3.5}}\put(11.6,2){\sf{\large{7 cm}}}\qbezier(8,1)(8,1)(11.5,3)\put(11.5,3){ \tt\large{D}}\put(11.3,1){\line(0,2){0.2}}\put(11.3,1.2){\line(2,0){0.2}}\end{picture}$

⠀⠀⠀$\rule{160}{1}$

✩ Diagonal Biscects Rectangle in two Equal Right Angle Triangle.

⠀

☢ $\underline{\textsf{By Pythagoras Theorem in \Delta BCD :}}$

$\dashrightarrow\sf\:\:(Diagonal)^2=(Length)^2+(Breadth)^2\\\\\\\dashrightarrow\sf\:\:(BD)^2=(BC)^2+(CD)^2\\\\\\\dashrightarrow\sf\:\:(BD)^2=(24\:cm)^2+(7\:cm)^2\\\\\\\dashrightarrow\sf\:\:(BD)^2=576\:cm^2+49\:cm^2\\\\\\\dashrightarrow\sf\:\:(BD)^2=625\:cm^2\\\\\\\dashrightarrow\sf\:\:BD=\sqrt{625\:cm^2}\\\\\\\dashrightarrow\sf\:\:BD=\sqrt{25\:cm \times 25\:cm}\\\\\\\dashrightarrow\:\:\underline{\boxed{\sf BD=25\:cm}}\qquad\bigg\lgroup\bf Diagonal\bigg\rgroup$

⠀

$\therefore\:\underline{\textsf{Hence, Length of Diagonal is C) \textbf{25 cm}}}.$

$\rule{180}{2}$

⠀⠀⠀⠀⠀⠀⠀⠀Shortcut Trick

$\begin{tabular}{|c |c | c|}\cline{1-3}l/b & b/l & h \\\cline{1-3}3 & 4 & 5 \\5 & 12 &13\\7 & 24&25 \\8 & 15&17\\\cline{1-3}\end{tabular}$

⠀

✩ As we know Rectangle will be Biscected by Diagonal in two Equal Right Angle Triangle.

✩ It will must follow Pythagorean Triplet.

✩ From Above we can see it will follow 7, 24 and 25. So , Longest Side will be Diagonal i.e. 25 cm.

⠀

$\therefore\:\underline{\textsf{Hence, Length of Diagonal is C) \textbf{25 cm}}}.$