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] In a \[\Delta PQR\], N is a point on PR such that \[QN\bot PR\]. If \[PN\times NR=Q{{N}^{2}}\] then\[\angle PQR=90{}^\circ \].

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

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: B

Solution :

In \[\Delta PQR\],

\[QN\bot PR\] and           \[PN\times RN=Q{{N}^{2}}\]

\[\frac{PN}{QN}=\frac{QN}{NR}\]

In\[\Delta PQN\]and \[\Delta RQN\],

\[\angle QNP=\angle QNR\]

\[\Delta QPN-\Delta RQN\]

[by SAS similarity]

\[\therefore \,\]\[\Delta QPN\]and \[\Delta RQN\]are equiangular.

\[\angle 1=\angle R\] and \[\angle 2=\angle P\]

\[\angle 1+\angle 2=\angle R+\angle P\]

\[\angle Q=\angle R+\angle P\]

Now, \[\angle Q+\angle R+\angle P=180{}^\circ\]

\[2\angle Q=180{}^\circ \left[ \angle Q=\angle R+\angle P \right]\]

\[\angle Q=90{}^\circ\]

Both Statements are true.