Answer

Verified

414.6k+ views

**Hint:**In this particular question use the concept that in a rhombus all the side lengths are equal and the adjacent sides w.r.t the particular sides of the given quadrilateral are perpendicular to each other and later on in the solution use the concept of Pythagoras theorem so use these concepts to reach the solution of the question.

__Complete step-by-step answer__:Proof –

As we see from the figure that in the given quadrilateral YMXN the adjacent sides are perpendicular to each other and opposite sides are parallel to each other but the adjacent side lengths are unequal.

So YMXN is a rectangle.

Now it is given that PQRS is a rhombus.

As we know that in a rhombus all the side lengths are equal.

Therefore, PQ = QR = RS = SP................. (1)

Now all the four triangles formed by the sides of the rhombus PQRS and the rectangle as shown in the figure are right angles.

So according to Pythagoras theorem, Hypotenuse square is equal to the sum of the square of base and perpendicular so we have,

$ \Rightarrow {\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{perpendicular}}} \right)^2} + {\left( {{\text{base}}} \right)^2}$

Now in right angle triangle PSY we have,

$ \Rightarrow {\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{perpendicular}}} \right)^2} + {\left( {{\text{base}}} \right)^2}$

$ \Rightarrow {\left( {{\text{SP}}} \right)^2} = {\left( {{\text{YS}}} \right)^2} + {\left( {PY} \right)^2}$.................. (2)

Now in right angle triangle SNR we have,

$ \Rightarrow {\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{perpendicular}}} \right)^2} + {\left( {{\text{base}}} \right)^2}$

$ \Rightarrow {\left( {{\text{RS}}} \right)^2} = {\left( {{\text{SN}}} \right)^2} + {\left( {{\text{NR}}} \right)^2}$.................. (3)

Now in right angle triangle RXQ we have,

$ \Rightarrow {\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{perpendicular}}} \right)^2} + {\left( {{\text{base}}} \right)^2}$

$ \Rightarrow {\left( {{\text{QR}}} \right)^2} = {\left( {{\text{QX}}} \right)^2} + {\left( {XR} \right)^2}$.................. (4)

Now add equation (2), (3) and (4) we have,

$ \Rightarrow {\left( {{\text{SP}}} \right)^2} + {\left( {{\text{RS}}} \right)^2} + {\left( {{\text{QR}}} \right)^2} = {\left( {{\text{YS}}} \right)^2} + {\left( {PY} \right)^2} + {\left( {{\text{SN}}} \right)^2} + {\left( {{\text{NR}}} \right)^2} + {\left( {{\text{QX}}} \right)^2} + {\left( {XR} \right)^2}$

Now from equation (1), PQ = QR = RS = SP we have,

$ \Rightarrow {\left( {{\text{PQ}}} \right)^2} + {\left( {{\text{PQ}}} \right)^2} + {\left( {{\text{PQ}}} \right)^2} = {\left( {{\text{YS}}} \right)^2} + {\left( {PY} \right)^2} + {\left( {{\text{SN}}} \right)^2} + {\left( {{\text{NR}}} \right)^2} + {\left( {{\text{QX}}} \right)^2} + {\left( {XR} \right)^2}$

$ \Rightarrow 3{\left( {{\text{PQ}}} \right)^2} = {\left( {{\text{YS}}} \right)^2} + {\left( {PY} \right)^2} + {\left( {{\text{SN}}} \right)^2} + {\left( {{\text{NR}}} \right)^2} + {\left( {{\text{QX}}} \right)^2} + {\left( {XR} \right)^2}$

Hence proved.

**Note**: Whenever we face such types of questions the key concept we have to remember is the Pythagoras theorem which is Hypotenuse square is equal to the sum of the square of base and perpendicular so apply this in all of the triangles except PMQ as above and add them we will get the required answer.

Recently Updated Pages

Differentiate between Shortterm and Longterm adapt class 1 biology CBSE

How do you find slope point slope slope intercept standard class 12 maths CBSE

How do you find B1 We know that B2B+2I3 class 12 maths CBSE

How do you integrate int dfracxsqrt x2 + 9 dx class 12 maths CBSE

How do you integrate int left dfracx2 1x + 1 right class 12 maths CBSE

How do you find the critical points of yx2sin x on class 12 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Define limiting molar conductivity Why does the conductivity class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Name 10 Living and Non living things class 9 biology CBSE

The Buddhist universities of Nalanda and Vikramshila class 7 social science CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE