Answer
Verified
410.4k+ 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.
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
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
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology 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
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE