Question

The ratio of two adjacent sides of a parallelogram is $ 2:3 $ and its perimeter is $ 50cm $ . Find its area if the altitude corresponding to the larger side is $ 10cm $ .

Answer 1

Hint: In the given question, we are provided with the ratio of two adjacent sides of parallelogram. The perimeter of parallelogram is $ 50cm $ . We first assume the two adjacent sides of the parallelogram in terms of a single variable with the help of the ratio given to us and then find the perimeter of the parallelogram by finding the sum of all the four sides. Then, solving the equation thus formed helps us to find the value of the variable and the sides of the parallelogram.

Complete step-by-step answer:
So, the ratio of the two adjacent sides of the parallelogram is $ 2:3 $ .
Let us assume the two adjacent sides of parallelogram to be $ 2x $ and $ 3x $ . This is also in accordance with the ratio of the sides provided to us in the question itself.
Also, we are given that the perimeter of the parallelogram is $ 50cm $ .
We know that the perimeter of a parallelogram is the sum of all the four sides. Also, we know that the opposite sides of parallelogram are equal. Hence, we get,
 $ \Rightarrow 2x + 3x + 2x + 3x = 50cm $
Adding up the like terms, we get,
 $ \Rightarrow 10x = 50cm $
Dividing both sides of the equation by $ 10 $ , we get,
 $ \Rightarrow x = 5cm $
So, substituting the value of x in the lengths of sides of parallelogram, we get,
 $ 2x = 2\left( {5cm} \right) = 10cm $ and $ 3x = 3\left( {5cm} \right) = 15cm $
So, we get the lengths of the two adjacent sides of the parallelogram as $ 10cm $ and $ 15cm $ respectively.
Now, we are given that the length of the altitude corresponding to the larger side of the parallelogram is $ 10cm $ .
We know that the formula for the area of the parallelogram is $ b \times h $ , where b is the base and h is the eight corresponding to that side or base of the parallelogram.
Hence, we get,
Area of the parallelogram $ = b \times h = \left( {15cm} \right) \times \left( {10cm} \right) = 150c{m^2} $
Hence, the area of parallelogram is $ 150c{m^2} $ .
So, the correct answer is “ $ 150c{m^2} $ ”.

Note: We must know how to solve an algebraic equation in order to solve the given problem. Algebraic equations can be solved in various ways. Method of transposition involves doing the exact same thing on both sides of an equation with the aim of bringing like terms together and isolating the variable or the unknown term in order to simplify the equation and finding the value of the required parameter. We must also remember the concepts of ratio and the formulae of area and perimeter of basic geometric figures.