Question

Assertion: It is always possible to draw a triangle whose sides measure 4 cm, 5 cm and 10 cm respectively.
Reason: In an isosceles ∆ABC with AB = AC, if BD and CE are bisectors of ∠B and ∠C respectively, then BD = CE.

(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Answer 1

(d) Assertion is false and Reason is true.
Reason (R):

$$\begin{gathered} AB=AC \\ \Rightarrow \angle C=\angle B \\ \Rightarrow \frac{1}{2}\angle C=\frac{1}{2}\angle B \\ \Rightarrow \angle BCE=\angle CBD \end{gathered}$$

$$\begin{gathered} \mathrm{In}△\mathrm{BDC}\mathrm{and}△\mathrm{CEB},\mathrm{we}\mathrm{have}: \\ \angle \mathrm{CBD}=\angle \mathrm{BCE}\left(\mathrm{Proved}\right) \\ \angle \mathrm{DCB}=\angle \mathrm{EBC}\left(\mathrm{Given}\right) \\ \mathrm{BC}=\mathrm{CB}\left(\mathrm{Common}\right) \\ \therefore △\mathrm{BDC}\cong \mathrm{CEB} \\ \Rightarrow \mathrm{BD}=\mathrm{CE}\left(\mathrm{CPCT}\right) \end{gathered}$$
Thus, the reason (R) is true.
The assertion (A) is false, as the sum of two sides of a triangle is always greater than the third side.
Here, $(4+5)<10$
So, the reason (R) is true, but the assertion (A) is false.