Answer
Verified
384k+ views
Hint: As we know that a trapezium is a quadrilateral having two parallel sides of unequal length and the other two sides are non- parallel. The parallel sides of a trapezium are called bases and the non parallel sides are called the legs. The area of the trapezium is $ \dfrac{1}{2} \times $ ( Sum of parallel sides) $ \times $ ( Distance between them).
Complete step by step solution:
First we will draw the diagram according to the questions:
In the above figure we can see that $ AB = 4 + \sqrt 5 $ and $ DC = 2 + x $ . The triangle $ ADX $ is a right angled triangle, so to find the value of AX we will apply the Pythagoras theorem.
We have $ AX = \sqrt {A{D^2} - D{X^2}} $ , by putting the values in the expression we have
$
AX = \sqrt {{7^2} - {2^2}} \\
\Rightarrow \sqrt {49 - 4} \;
$ .
It gives us the value $ AX = \sqrt {45} = 3\sqrt {5.} $
We have been given that the area of the trapezium is $ 15(15 + \sqrt 2 )c{m^2}. $ Now the area of the trapezium is $ \dfrac{1}{2} \times $ ( Sum of parallel sides) $ \times $ ( Distance between them).
So we can write
$ 15(\sqrt 5 + 2) = \dfrac{1}{2} \times \left( {4 + \sqrt 5 + 2 + x} \right) \times 3\sqrt 5 $ .
We will solve it now by isolating the term $ x $ .
Therefore: $ \left( {6 + \sqrt 5 + x} \right)3\sqrt 5 = 2 \times 15\left( {\sqrt 5 + 2} \right) $ . Now we will transfer the $ 3\sqrt 5 $ to the right hand side of the equation,
$ x + 6 + \sqrt 5 = 30\dfrac{{\sqrt 5 + 2}}{{3\sqrt 5 }} $ .
Taking the common factor out $ x + 6 + \sqrt 5 = 10\left( {1 + \dfrac{2}{5}\sqrt 5 } \right) $ , transfer all the terms except the variable to the right hand side
$
x = 10 + \dfrac{{20}}{5} \times \sqrt 5 - 6 - \sqrt 5 \\
\Rightarrow 4 + 3\sqrt 5 \;
$ .
Hence the required answer is $ 4 + 3\sqrt 5 . $
So, the correct answer is “ $ 4 + 3\sqrt 5 $ ”.
Note: Before solving this kind of question we should know all the properties and the formulas of the trapezium. We should note that the question says that the AX is perpendicular so that makes the angle $ {90^ \circ } $ , hence it forms the right angled triangle. We should be careful while solving to avoid the calculation mistakes.
Complete step by step solution:
First we will draw the diagram according to the questions:
In the above figure we can see that $ AB = 4 + \sqrt 5 $ and $ DC = 2 + x $ . The triangle $ ADX $ is a right angled triangle, so to find the value of AX we will apply the Pythagoras theorem.
We have $ AX = \sqrt {A{D^2} - D{X^2}} $ , by putting the values in the expression we have
$
AX = \sqrt {{7^2} - {2^2}} \\
\Rightarrow \sqrt {49 - 4} \;
$ .
It gives us the value $ AX = \sqrt {45} = 3\sqrt {5.} $
We have been given that the area of the trapezium is $ 15(15 + \sqrt 2 )c{m^2}. $ Now the area of the trapezium is $ \dfrac{1}{2} \times $ ( Sum of parallel sides) $ \times $ ( Distance between them).
So we can write
$ 15(\sqrt 5 + 2) = \dfrac{1}{2} \times \left( {4 + \sqrt 5 + 2 + x} \right) \times 3\sqrt 5 $ .
We will solve it now by isolating the term $ x $ .
Therefore: $ \left( {6 + \sqrt 5 + x} \right)3\sqrt 5 = 2 \times 15\left( {\sqrt 5 + 2} \right) $ . Now we will transfer the $ 3\sqrt 5 $ to the right hand side of the equation,
$ x + 6 + \sqrt 5 = 30\dfrac{{\sqrt 5 + 2}}{{3\sqrt 5 }} $ .
Taking the common factor out $ x + 6 + \sqrt 5 = 10\left( {1 + \dfrac{2}{5}\sqrt 5 } \right) $ , transfer all the terms except the variable to the right hand side
$
x = 10 + \dfrac{{20}}{5} \times \sqrt 5 - 6 - \sqrt 5 \\
\Rightarrow 4 + 3\sqrt 5 \;
$ .
Hence the required answer is $ 4 + 3\sqrt 5 . $
So, the correct answer is “ $ 4 + 3\sqrt 5 $ ”.
Note: Before solving this kind of question we should know all the properties and the formulas of the trapezium. We should note that the question says that the AX is perpendicular so that makes the angle $ {90^ \circ } $ , hence it forms the right angled triangle. We should be careful while solving to avoid the calculation mistakes.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Which are the Top 10 Largest Countries of the World?
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE