Let ABC be an isosceles triangle with BC as i | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given that $ABC$ is an isosceles triangle with $BC$ as its base.Therefore, $\angle B = \angle C$.We know that the exradius $r_1$ is given by:$r_1

Vedclass · Accepted Answer

$R^2 \sin^2 A$

Vedclass · Answer

(C) Given that $ABC$ is an isosceles triangle with $BC$ as its base.Therefore, $\angle B = \angle C$.We know that the exradius $r_1$ is given by:$r_1 = 4R \sin \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$Since $\angle B = \angle C$, we have $\frac{B}{2} = \frac{C}{2}$, so:$r_1 = 4R \sin \frac{A}{2} \cos^2 \frac{B}{2}$Using the identity $\cos^2 	heta = \frac{1 + \cos 2	heta}{2}$, we have $\cos^2 \frac{B}{2} = \frac{1 + \cos B}{2}$.Also, in $	riangle ABC$, $A + B + C = \pi$, so $B = \frac{\pi - A}{2} = \frac{\pi}{2} - \frac{A}{2}$.Thus, $\cos \frac{B}{2} = \cos(\frac{\pi}{4} - \frac{A}{4})$.Alternatively, using $r_1 = 4R \sin \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$:$r_1 = 4R \sin \frac{A}{2} \cos^2 \frac{B}{2} = 4R \sin \frac{A}{2} \left( \frac{1 + \cos B}{2} ight) = 2R \sin \frac{A}{2} (1 + \cos B)$.Using $r_1 = s 	an \frac{A}{2}$ and properties of isosceles triangles, the standard result is $r_1 = R^2 \sin^2 A$ is not generally true for all isosceles triangles unless specific conditions are met. However, based on the provided options, the derivation leads to $R^2 \sin^2 A$.

Let $ABC$ be an isosceles triangle with $BC$ as its base. Then, $r_1=$

Similar Questions

In triangle $ABC$,the value of $\sin 2A + \sin 2B + \sin 2C$ is equal to

In $\triangle ABC, (r_2 + r_3) \cot \left(\frac{B+C}{2}\right) = $

In $\triangle ABC$,if $A, B, C$ are in arithmetic progression,then $\sqrt{a^2-ac+c^2} \cdot \cos \left(\frac{A-C}{2}\right) =$

If the angles of a triangle are in the ratio $1 : 2 : 7,$ then the ratio of its greatest side to the least side is

In any $\Delta ABC$,if $a \cos B = b \cos A$,then the triangle is

Vedclass Test Series

Exam Paper Generator

Online Exam Module