In the figure, ABCDE is a pentagon with \[BE||CD\] and \[BC||DE\]. BC is perpendicular to CD. \[AB=5cm\], \[AE=5cm\], \[BE=7cm\], \[BC=x-y\] and \[CD=x+y\]. If the perimeter of ABCDE is 27cm, find the value of ‘x’ and ‘y’, given x, y = 0.

Question

In the figure, ABCDE is a pentagon with \[BE||CD\] and \[BC||DE\]. BC is perpendicular to CD. \[AB=5cm\], \[AE=5cm\], \[BE=7cm\], \[BC=x-y\] and \[CD=x+y\]. If the perimeter of ABCDE is 27cm, find the value of ‘x’ and ‘y’, given x, y = 0.

Vedantu Content Team · Accepted Answer

Hint: We are given a question to find the value of ‘x’ and ‘y’ in the given pentagon. We are given some of the sides and we have to compute the lengths to find the lengths ‘x’ and ‘y’. We will first have the figure, then based on the given measurements, we will form equations. We have the equations as, $$3x-y=17$$ and $$x+y=7$$. On solving these equations, we will get the value of ‘x’ and ‘y’. Hence, we will have the required values. Complete step-by-step answer:According to the given question, we have to find the values of the given variables ‘x’ and ‘y’. We will first draw the diagram with the given measurements and we have,\n    \n    \n    \n    \n   We are given a pentagon ABCDE, and that the sides $$BE||CD$$ and $$BC||DE$$.So, we have the lengths of BC and BE same as well as the lengths of BE and CD are same.We are also given that the perimeter of the given pentagon is 27cm, that is,$$5+(x-y)+(x+y)+(x-y)+5=27$$We will find the equation by the solving the above expression, we get,$$\Rightarrow 10+x-y+x+y+x-y=27$$Cancelling out the common terms and we get,$$\Rightarrow 10+3x-y=27$$$$\Rightarrow 3x-y=17$$----(1)Also, since $$BE||CD$$, we have,$$x+y=7$$----(2)We will now solve the equation (1) and (2), we get,Adding equation (1) and (2), we get the new expression as,$$4x=17+7$$$$\Rightarrow 4x=24$$$$\Rightarrow x=6$$So, we get the value of $$x=6$$, substituting this value in the equation (2), we get,$$6+y=7$$$$\Rightarrow y=7-6=1$$Therefore, the value of $$x=6$$ and $$y=1$$.Note: The pentagon’s base can be said to be similar to a rectangle, where the opposite sides are equal. Using this property, we were able to solve for the values of ‘x’ and ‘y’. So, having obtained the values, we can find the lengths of the sides, $$x-y=6-1=5$$ and $$x+y=6+1=7$$, that is, $$BC=5cm$$ and $$CD=7cm$$.