question_answer

 

Directions: Each of these questions contains two statements:

Assertion [A] and Reason [R]. Each of these questions also has four alternative choices, any one of which is the correct answer. You have to select one of the codes [a], [b], [c] and [d] given below.

Assertion [A] \[\Delta ABC\] is a right triangle right angled at B. Let D and E be any points on AB and BC respectively.

Then, \[A{{E}^{2}}+C{{D}^{2}}=A{{C}^{2}}+D{{E}^{2}}\].

Reason [R] In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

 

A) A is true, R is true; R is a correct explanation for A.

B) A is true, R is true; R is not a correct explanation for A.

C) A is true; R is False.

D) A is false; R is true.

Correct Answer: A

Solution :

Since, \[\Delta ABE\]is right triangle right-angled at B. \[A{{E}^{2}}=A{{B}^{2}}+B{{E}^{2}}\]                           ...(i) Again, \[\Delta DBC\]is right triangle right angled at B. \[C{{D}^{2}}=B{{D}^{2}}+B{{C}^{2}}\]                     ...(ii) Adding Eqs. (i) and (ii), we get \[A{{E}^{2}}+C{{D}^{2}}=\left( A{{B}^{2}}+B{{E}^{2}} \right)+\left( B{{D}^{2}}+B{{C}^{2}} \right)\] \[\Rightarrow A{{E}^{2}}+C{{D}^{2}}=\left( A{{B}^{2}}+B{{C}^{2}} \right)+\left( B{{E}^{2}}+B{{D}^{2}} \right)\]Using Pythagoras theroem in \[\Delta ABC\]and \[\Delta DBE\], we have \[A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}\] and   \[D{{E}^{2}}=B{{E}^{2}}+B{{D}^{2}}\] \[\therefore \,\,\,A{{E}^{2}}+C{{D}^{2}}=A{{C}^{2}}+D{{E}^{2}}\] Hence, \[A{{E}^{2}}+C{{D}^{2}}=A{{C}^{2}}+D{{E}^{2}}\] Both Statement I and II are true and Statement II is the correct explanation of Statement I.