question_answer

Assertion (A): ABC is a triangle in which \[\text{AB}=\text{AC}\] and D is a point on AC such that \[B{{C}^{2}}=AC\times CD\]. Then \[\Delta ABC\tilde{\ }\Delta BDC\]by SAS similarity criterion.

Reason (R): If two angles of one triangle are respectively equal to the two angles of another triangle, then the two triangles are similar. This is known as SAS similarity criterion.

A) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

B) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

C) Assertion (A) is true but reason (R) is false.

D) Assertion (A) is false but reason (R) is true.

Correct Answer: C

Solution :

[c] Clearly, reason is false as it is AA similarity criterion.

We are given that \[B{{C}^{2}}=AC\times CD\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{BC}{CD}=\frac{AC}{BC}\]                ..(1)

In \[\Delta ABC\] and \[\Delta BDC,\]we have

\[\frac{AC}{BC}=\frac{BC}{CD}\]

and       \[\angle BCA=\angle DCB\]                   (Common)

\[\therefore \,\,\,\Delta ABC\tilde{\ }\Delta BDC\]

(By SAS similarity criterion)

\[\therefore \] Assertion: True; Reason: False.