The perimeter of a rectangular room is 34m and the length of a diagona

1.	The perimeter of a rectangular room is 34m and the length of a diagonal is 13m. The dimensions of the room are ……………A) 17m and 7m B) 17m and 5m C) 9m and 2m D) 15m and 2m
Answer» 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

The perimeter of a rectangular room is 34m and the length of a diagonal is 13m. The dimensions of the room are ……………A) 17m and 7m B) 17m and 5m C) 9m and 2m D) 15m and 2m

Answer»

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