Statement A (Assertion) : A B C is a triangle in which A B=A C an… - Vidyadip

MCQ

Statement A (Assertion) : $A B C$ is a triangle in which $A B=A C$ and $D$ is a point on $A C$ such that $B C^2=A C \times C D$. Then, $\triangle A B C \sim \triangle B D C$ by SAS similarity criterion.
Statement R (Reason) : If two angles of one triangle are respectively equal to the two angles of another triangle, then the two triangles are similar. This is knownas SAS similaritycriterion.

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 and reason $(R)$ is not the correct explanation of assertion (A).
✓
Assertion (A) is true but reason $(R)$ is false.
D
Assertion (A) is false but reason (R) is true.

✓

Answer

Correct option: C.

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

(c) : Clearly, Reason is false as it is AA similarity criterion.

We are given that $B C^2=A C \times C D$
$
\Rightarrow \frac{B C}{C D}=\frac{A C}{B C}\ldots(i)
$
In $\triangle A B C$ and $\triangle B D C$, we have
$
\frac{A C}{B C}=\frac{B C}{C D}(From (i)
$
and $\angle B C A=\angle D C B$(Common)
$\therefore \quad \triangle A B C \sim \triangle B D C$(By SAS similarity criterion)
$\therefore \quad$ Assertion is true.

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