The perimeter of a rectangular room is 34m and the length of a diagonal is 13m. The dimensions of the room are ……………

Correct option is (B) 17m and 5m

Let \(l\;and\;b\) are parameters of rectangular room.

\(\therefore2(l+b)=34\)        \((\because\) Perimeter = 34 m (given))

\(\Rightarrow l+b=\frac{34}2=17\)   _____________(1)

\(\because\) Length of diagonal = 13

\(\Rightarrow\sqrt{l^2+b^2}=13\)

\(\Rightarrow l^2+b^2=13^2=169\)   _____________(2)

Now, \((l+b)^2=l^2+b^2+2lb\)

\(\Rightarrow17^2=169+2lb\)      (From (1) & (2))

\(\Rightarrow2lb=289-169\)

\(\Rightarrow lb=\frac{120}2=60\)   _____________(3)

Now, \((l-b)^2=(l+b)^2-4lb\)

\(=17^2-4\times60\)          (From (1) & (3))

= 289 - 240

= 49 \(=7^2\)

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

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

\((l+b)+(l-b)=17+7\)

\(\Rightarrow2l=24\)

\(\Rightarrow l\) = 12 m

Put \(l=12\) in equation (1), we get

12+b = 17

\(\Rightarrow\) b = 17 - 12 = 5 m

Hence, dimensions of the room are 17m and 5m.

Correct option is B) 17m and 5m