Answer
457.5k+ views
Hint: Apply Pythagoras theorem here while solving.
Pictorial representation of the given problem is shown below.
Two points D and E are on the sides AC and AB.
We have to prove $A{E^2} + B{D^2} = A{B^2} + D{E^2}$
Join the point D with B and E and the point E with A.
Proof: -
Using Pythagoras Theorem,
We know that
${\text{Hypotenuse}}{{\text{e}}^2} = {\text{Perpendicular}}{{\text{r}}^2} + {\text{Bas}}{{\text{e}}^2}$
In $\Delta ACE,$
$A{E^2} = A{C^2} + C{E^2}.........\left( 1 \right)$
In $\Delta BCD,$
$B{D^2} = D{C^2} + B{C^2}.........\left( 2 \right)$
In $\Delta ABC,$
$A{B^2} = A{C^2} + B{C^2}.........\left( 3 \right)$
In $\Delta DCE,$
$D{E^2} = D{C^2} + C{E^2}.........\left( 4 \right)$
Now, consider L.H.S which is$A{E^2} + B{D^2}$
From equation (1) and (2)
$A{E^2} + B{D^2} = A{C^2} + C{E^2} + D{C^2} + B{C^2}.........\left( 5 \right)$
Now, consider R.H.S which is$A{B^2} + D{E^2}$
From equation (3) and (4)
$A{B^2} + D{E^2} = A{C^2} + B{C^2} + D{C^2} + C{E^2}.........\left( 6 \right)$
Now from equation (5) and (6) it is clear that L.H.S = R.H.S
Therefore $A{E^2} + B{D^2} = A{B^2} + D{E^2}$
Hence Proved.
Note:In such types of questions first draw the pictorial representation of the given problem then, apply Pythagoras Theorem in all different right angle triangles which is right angled at C, then find out the value of L.H.S and R.H.S separately, then we see that both are equal which is the required result.
Pictorial representation of the given problem is shown below.
![seo images](https://www.vedantu.com/question-sets/0d50234b-3946-4305-9d43-b3d348736d393547427329444669027.png)
Two points D and E are on the sides AC and AB.
We have to prove $A{E^2} + B{D^2} = A{B^2} + D{E^2}$
Join the point D with B and E and the point E with A.
Proof: -
Using Pythagoras Theorem,
We know that
${\text{Hypotenuse}}{{\text{e}}^2} = {\text{Perpendicular}}{{\text{r}}^2} + {\text{Bas}}{{\text{e}}^2}$
In $\Delta ACE,$
$A{E^2} = A{C^2} + C{E^2}.........\left( 1 \right)$
In $\Delta BCD,$
$B{D^2} = D{C^2} + B{C^2}.........\left( 2 \right)$
In $\Delta ABC,$
$A{B^2} = A{C^2} + B{C^2}.........\left( 3 \right)$
In $\Delta DCE,$
$D{E^2} = D{C^2} + C{E^2}.........\left( 4 \right)$
Now, consider L.H.S which is$A{E^2} + B{D^2}$
From equation (1) and (2)
$A{E^2} + B{D^2} = A{C^2} + C{E^2} + D{C^2} + B{C^2}.........\left( 5 \right)$
Now, consider R.H.S which is$A{B^2} + D{E^2}$
From equation (3) and (4)
$A{B^2} + D{E^2} = A{C^2} + B{C^2} + D{C^2} + C{E^2}.........\left( 6 \right)$
Now from equation (5) and (6) it is clear that L.H.S = R.H.S
Therefore $A{E^2} + B{D^2} = A{B^2} + D{E^2}$
Hence Proved.
Note:In such types of questions first draw the pictorial representation of the given problem then, apply Pythagoras Theorem in all different right angle triangles which is right angled at C, then find out the value of L.H.S and R.H.S separately, then we see that both are equal which is the required result.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)