Answer

Verified

389.1k+ views

**Hint:**Analyze the situation with a diagram. Take a random point O inside the square and join it with every vertex. Draw two lines passing through O and parallel to the sides AB and BC respectively. Consider four right angled triangles each to determine the values of OA, OB, OC and OD respectively in terms of some variables. Then verify the result $O{A^2} + O{C^2} = O{B^2} + O{D^2}$.

**Complete step by step answer:**

According to the question, a rectangle ABCD is said to have any interior point O. We have to prove that $O{A^2} + O{C^2} = O{B^2} + O{D^2}$.

Consider the rectangle ABCD shown below with a point O lying inside it.

OA, OB, OC and OD are the lines joining the vertex of the square and point O. We have also drawn EG and FH parallel to the sides of the square and passing through point O.

From this we can conclude that AD, FH and BC are parallel and equal. Similarly AB, EG and DC are also parallel and equal.

So let $AF = OE = DH = a$.

Similarly we will assume some variable for other sides also as shown below:

$

\Rightarrow FB = OG = HC = b \\

\Rightarrow AE = OF = BG = c \\

\Rightarrow ED = OH = GC = d \\

$

To find the values of OA, OB, OC and OD, we’ll consider right angled triangles.

So in right angled triangle $AOF$, we have:

$ \Rightarrow O{A^2} = O{F^2} + A{F^2} = {c^2} + {a^2}{\text{ }}.....{\text{(1)}}$

Similarly in triangle $BOG$, we have:

$ \Rightarrow O{B^2} = O{G^2} + B{G^2} = {b^2} + {c^2}{\text{ }}.....{\text{(2)}}$

In triangle $COH$, we have:

$ \Rightarrow O{C^2} = O{H^2} + H{C^2} = {d^2} + {b^2}{\text{ }}.....{\text{(3)}}$

And in triangle $DOE$, we have:

$ \Rightarrow O{D^2} = O{E^2} + E{D^2} = {a^2} + {d^2}{\text{ }}.....{\text{(4)}}$

Now adding equation (1) and (3), we’ll get:

$

\Rightarrow O{A^2} + O{C^2} = {c^2} + {a^2} + {d^2} + {b^2} \\

\Rightarrow O{A^2} + O{C^2} = {a^2} + {b^2} + {c^2} + {d^2}{\text{ }}.....{\text{(5)}} \\

$

And adding equation (2) and (4), we’ll get:

$

\Rightarrow O{B^2} + O{D^2} = {b^2} + {c^2} + {a^2} + {d^2} \\

\Rightarrow O{B^2} + O{D^2} = {a^2} + {b^2} + {c^2} + {d^2}{\text{ }}.....{\text{(6)}} \\

$

On comparing equation (5) and (6), we can say that:

$ \Rightarrow O{A^2} + O{C^2} = O{B^2} + O{D^2}$

Hence this is proved.

Further we have to calculate the length of OD such that OA, OB and OC are 3 cm, 4 cm and 5 cm respectively.

So using the same result:

$ \Rightarrow O{A^2} + O{C^2} = O{B^2} + O{D^2}$

Putting the values, we’ll get:

$

\Rightarrow {3^2} + {5^2} = {4^2} + O{D^2} \\

\Rightarrow 16 + O{D^2} = 9 + 25 = 34 \\

\Rightarrow O{D^2} = 18 \\

\Rightarrow OD = \sqrt {18} = 3\sqrt 2 \\

$

**Thus the length of OD is $3\sqrt 2 $ cm.**

**Note:**Although we have proved the above result for rectangles, this will hold true for squares also. Since we have only used the property of square that it’s opposite sides are parallel and equal and all of its angles are ${90^ \circ }$ and this property is also followed by square, thus the result will be equally valid for squares.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

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

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Change the following sentences into negative and interrogative class 10 english CBSE