Answer
Verified
376.5k+ views
Hint: First of all find the length of the side CD. To find the length of the side CE consider ABCE as a rectangle and use the fact that the opposite sides of a rectangle are equal and use the relations CE = AB and BC = AE. To find the length of the side DE consider AD as hypotenuse (H), DE as the base (B) and AE as the perpendicular (P) of the right angle triangle AED. Use the Pythagoras theorem given as ${{H}^{2}}={{P}^{2}}+{{B}^{2}}$ to find the value of DE. Add CE and DE to find the length of CD and finally take the sum of the length of all the sides of the trapezium ABCD to get the answer.
Complete step by step solution:
Here we have been provided with a trapezium with the following measurements shown below. We are asked to calculate the perimeter of the trapezium. First we need to find the length of the fourth side that is CD.
Now, clearly we can consider ABCE as a rectangle and we know that the opposite sides of a rectangle are equal so we have the following relations: -
$\Rightarrow $ AE = BC = 15 ……. (1)
$\Rightarrow $ AB = CE = 20 ……. (2)
Now, in the right angle triangle AED we have AD as hypotenuse (H), DE as the base (B) and AE as the perpendicular (P). Therefore, using the Pythagoras theorem given as ${{H}^{2}}={{P}^{2}}+{{B}^{2}}$ we get,
$\Rightarrow {{\left( AD \right)}^{2}}={{\left( AE \right)}^{2}}+{{\left( DE \right)}^{2}}$
Substituting the values AD = 17 and AE = 15 {relation (1)} we get,
$\begin{align}
& \Rightarrow {{\left( 17 \right)}^{2}}={{\left( 15 \right)}^{2}}+{{\left( DE \right)}^{2}} \\
& \Rightarrow {{\left( DE \right)}^{2}}=289-225 \\
& \Rightarrow {{\left( DE \right)}^{2}}=64 \\
\end{align}$
Taking square root both the sides we get,
$\Rightarrow $ DE = 8
Therefore the total length of the side CD is sum of the length of the sides CE and DE, so we get,
$\Rightarrow $ CD = CE + DE
Substituting the obtained value of DE and using relation (2) for the value of CE we get,
$\Rightarrow $ CD = 20 + 8
$\Rightarrow $ CD = 28
Now, the perimeter of the trapezium ABCD is the sum of the length of all the sides of the trapezium, so we have,
$\Rightarrow $ Perimeter = AB + BC + CD + DA
$\Rightarrow $ Perimeter = 20 + 15 + 28 + 17
$\therefore $ Perimeter = 80
So, the correct answer is “Option d”.
Note: You must remember the properties of certain special quadrilaterals like parallelogram, square, rectangle, rhombus etc. as any of their properties may be used in the questions where we need to find the area or the perimeter of a figure. Also remember the formula for the area of a trapezium given as $\dfrac{1}{2}\times $ sum of parallel sides $\times $ perpendicular distance between the parallel sides.
Complete step by step solution:
Here we have been provided with a trapezium with the following measurements shown below. We are asked to calculate the perimeter of the trapezium. First we need to find the length of the fourth side that is CD.
Now, clearly we can consider ABCE as a rectangle and we know that the opposite sides of a rectangle are equal so we have the following relations: -
$\Rightarrow $ AE = BC = 15 ……. (1)
$\Rightarrow $ AB = CE = 20 ……. (2)
Now, in the right angle triangle AED we have AD as hypotenuse (H), DE as the base (B) and AE as the perpendicular (P). Therefore, using the Pythagoras theorem given as ${{H}^{2}}={{P}^{2}}+{{B}^{2}}$ we get,
$\Rightarrow {{\left( AD \right)}^{2}}={{\left( AE \right)}^{2}}+{{\left( DE \right)}^{2}}$
Substituting the values AD = 17 and AE = 15 {relation (1)} we get,
$\begin{align}
& \Rightarrow {{\left( 17 \right)}^{2}}={{\left( 15 \right)}^{2}}+{{\left( DE \right)}^{2}} \\
& \Rightarrow {{\left( DE \right)}^{2}}=289-225 \\
& \Rightarrow {{\left( DE \right)}^{2}}=64 \\
\end{align}$
Taking square root both the sides we get,
$\Rightarrow $ DE = 8
Therefore the total length of the side CD is sum of the length of the sides CE and DE, so we get,
$\Rightarrow $ CD = CE + DE
Substituting the obtained value of DE and using relation (2) for the value of CE we get,
$\Rightarrow $ CD = 20 + 8
$\Rightarrow $ CD = 28
Now, the perimeter of the trapezium ABCD is the sum of the length of all the sides of the trapezium, so we have,
$\Rightarrow $ Perimeter = AB + BC + CD + DA
$\Rightarrow $ Perimeter = 20 + 15 + 28 + 17
$\therefore $ Perimeter = 80
So, the correct answer is “Option d”.
Note: You must remember the properties of certain special quadrilaterals like parallelogram, square, rectangle, rhombus etc. as any of their properties may be used in the questions where we need to find the area or the perimeter of a figure. Also remember the formula for the area of a trapezium given as $\dfrac{1}{2}\times $ sum of parallel sides $\times $ perpendicular distance between the parallel sides.
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
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE