Mathematics[[1]] in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of two sides.

[[1]] in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of two sides.


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    Solution:

    A right-angled triangle ABC which is right-angled at B
    To prove:  A C 2 =A B 2 +B C 2  
    Construction: Draw a perpendicular BD from B to AC
    IMG_256
    Proof:-
    In triangle ABC and ABD
    ADB=ABC= 90 (Given) DAB=BAC=(common angle) ΔABCΔABD using AA similarity criteria  
    AA similarity criterion :- It states that if two triangles have two pairs of congruent angles, then the triangles are similar.
    Now, Applying properties of similar triangles, we have corresponding sides proportional to each other.
    AD AB = AB AC A B 2 =AD×ACeqn.1  
    Similarly, ΔABCΔBDC(using AA similarity criteria)  
    Again, Applying properties of similar triangle, we have corresponding sides proportional to each other
    BC CD = AC BC  
    B C 2 =CD×ACeqn.2  
    Adding equation (1) and (2), we get
    A B 2 +B C 2 =AD×AC+CD×AC  
    Taking AC common, we get
    A B 2 +B C 2 =AC(AD+CD)eqn.3  
    And from the figure we know that AD + CD = AC
    After changing the obtained value of AC above in equation (3), we get
    A B 2 +B C 2 =AC×AC=A C 2  
    in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of two sides.
     
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