question_answer

Assertion (A): ABC is an isosceles, right triangle, right angled at C. Then \[A{{B}^{2}}=3A{{C}^{2}}\]

Reason (R): In an isosceles triangle ABC if \[\text{AC}=\text{BC}\] and \[A{{B}^{2}}=2A{{C}^{2}},\]then \[\angle C=90{}^\circ \]

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: D

Solution :

[d] If assertion is incorrect, but reason is correct.

In right angled \[\Delta ABC,\]

\[A{{B}^{2}}=A{{C}^{2}}+B{{C}^{2}}\]

(By Pythagoras Theorem)

\[=A{{C}^{2}}+A{{C}^{2}}=2A{{C}^{2}}\]          \[[BC=AC]\]

\[A{{B}^{2}}=2A{{C}^{2}}\]

Assertion is false.

Again since,

\[A{{B}^{2}}=2A{{C}^{2}}=A{{C}^{2}}+A{{C}^{2}}=A{{C}^{2}}+B{{C}^{2}}\]

[\[AC=BC,\] (Given)]

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

(By converse of Pythagoras Theorem)

Reason is true.