If the perimeter of a rectangular room is 34 and the length of the diagonal is 13, then the dimensions of the room are ………… 

Correct option is (C) 12, 5

Let length & breadth of the rectangular room are \(l\;and\;b\) respectively.

\(\because\) Perimeter of a rectangular room is 34.

i.e., \(2(l+b)\) = 34

\(\Rightarrow\) \(l+b=\frac{34}2\) = 17

\(\Rightarrow\) \(l+b=17\)   _________(1)

\(\because\) Length of the diagonal is 13.

\(\therefore\) \(\sqrt{l^2+b^2}=13\)

\(\Rightarrow\) \(l^2+b^2=13^2\)

\(\Rightarrow\) \(l^2+b^2=169\)    _________(2)

\(\because\) \((l+b)^2=l^2+b^2+2lb\)

\(\Rightarrow\) \(17^2=169+2lb\)    (From (1) and (2))

\(\Rightarrow\) \(2lb=17^2-169\)

= 289 - 169

\(\Rightarrow\) \(2lb=120\)    _________(3)

\(\therefore\) \((l-b)^2=l^2+b^2-2lb\)

= 169 - 120    (From (3))

= 49 \(=7^2\)

\(\Rightarrow\) \(l-b=7\)    _________(4)

By adding (1) & (4), we get

\(2l\) = 17+7 = 24

\(\Rightarrow l=\frac{24}2=12\)

\(\therefore\) b = 17+b

\(\Rightarrow\) b = 17 - 12 = 5

Hence, the dimensions of the room are \(l=12\) & b = 5.

Correct option is C) 12, 5