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SAS Criteria for Congruency

In the figure...

Question

In the figure $A B = B C$ and AD is perpendicular to CD. Then,

A C^{2} = 2. A D . D C .

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A C^{2} = 2. D B . D C .

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A C^{2} = 2. B C . D C .

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A C^{2} = 2. B C . A D .

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Solution

The correct option is B $A C^{2} = 2. B C . D C .$
In $△ A D B$ ,
$A B^{2} = A D^{2} + D B^{2}$ (By Pythagoras Theorem)
or $A D^{2} = A B^{2} - D B^{2}$
In $△ A D C$
$A C^{2} = A D^{2} + D C^{2}$ (By Pythagoras Theorem)
Now, $A C^{2} = A B^{2} - D B^{2} + (D B + B C)^{2}$
$A C^{2} = A B^{2} - D B^{2} + D B^{2} + B C^{2} + 2 D B . B C$
$A C^{2} = B C^{2} + B C^{2} + 2 D B . B C$ (Since, $A B = B C$ )
$A C^{2} = 2 B C^{2} + 2 D B . B C$
$A C^{2} = 2 B C . (B C + D B)$
$A C^{2} = 2 B C . D C$

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If $A D ⊥ B C$ , prove that ${A B}^{2} + {C D}^{2} = {B D}^{2} + {A C}^{2}$ . [2 MARKS]

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