question_answer

The ratio of two smaller sides of a right-angled triangle is 4 : 3, A rectangle is on the largest side of the triangle in such a way that largest side will be the length of the rectangle. The breadth of rectangle is four fifth of its length. Find the length of shortest side of triangle if the perimetre of rectangle is 1.8 m.

A) 60 cm       

B)               40 cm

C) 15 cm       

D)               30 cm

E) None of these

Correct Answer: D

Solution :

Explanation Let the small sides are 4x and 3x. From the figure \[{{\operatorname{BC}}^{2}}=A{{B}^{2}}=A{{C}^{2}}\]   (Pythagoras theorem) \[\Rightarrow \,\,\,\,{{\operatorname{BC}}^{2}}= {{\left( 3x \right)}^{2}}+ 4{{x}^{2}}\] \[\Rightarrow \,\,\,{{\operatorname{BC}}^{2}}= 9{{x}^{2}}+ 16{{x}^{2}}\] \[\Rightarrow \,\,\,{{\operatorname{BC}}^{2}}= 25{{x}^{2}}BC = 5x\] According to question, \[\operatorname{Length} of rectangle = 5x\] Breadth of rectangle \[\Rightarrow \,\,\,\frac{4}{5}\times 5x\,\,=\,\,4x\] Perimeter of rectangle \[= \,2\left( 5x +4x \right) = 18x\] \[\Rightarrow \,\,\,\,18x = 180 cm\] \[\Rightarrow \,\,\,x= 10\,\,cm\] The length of shortest side \[= 3 \times  10 = 30 cm.\]