If p_1, p_2, p_3 are altitudes of a triangle | vedclass

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

Question

(A) We know that the area of the triangle $\Delta = \frac{1}{2} a p_1 = \frac{1}{2} b p_2 = \frac{1}{2} c p_3$. Thus,$p_1 = \frac{2\Delta}{a}$,$p_2 =

Vedclass · Accepted Answer

$\frac{1}{R}$

Vedclass · Answer

(A) We know that the area of the triangle $\Delta = \frac{1}{2} a p_1 = \frac{1}{2} b p_2 = \frac{1}{2} c p_3$. Thus,$p_1 = \frac{2\Delta}{a}$,$p_2 = \frac{2\Delta}{b}$,and $p_3 = \frac{2\Delta}{c}$. Substituting these into the expression: $\frac{\cos A}{p_1} + \frac{\cos B}{p_2} + \frac{\cos C}{p_3} = \frac{a \cos A}{2\Delta} + \frac{b \cos B}{2\Delta} + \frac{c \cos C}{2\Delta}$. Using the sine rule $a = 2R \sin A$,$b = 2R \sin B$,$c = 2R \sin C$: $= \frac{2R \sin A \cos A + 2R \sin B \cos B + 2R \sin C \cos C}{2\Delta}$ $= \frac{R(\sin 2A + \sin 2B + \sin 2C)}{2\Delta}$. In any triangle,$\sin 2A + \sin 2B + \sin 2C = 4 \sin A \sin B \sin C$. Also,$\Delta = 2R^2 \sin A \sin B \sin C$,so $\sin A \sin B \sin C = \frac{\Delta}{2R^2}$. Substituting this: $= \frac{R(4 \cdot \frac{\Delta}{2R^2})}{2\Delta} = \frac{2R \Delta / R^2}{2\Delta} = \frac{1}{R}$.

If $p_1, p_2, p_3$ are altitudes of a triangle $ABC$ from the vertices $A, B, C$ respectively and if $\Delta$ is the area of the triangle,$S$ is the semi-perimeter of the triangle,then find the value of $\frac{\cos A}{p_1} + \frac{\cos B}{p_2} + \frac{\cos C}{p_3}$.

Similar Questions

In $\Delta ABC$, if $a^{2} \cos^{2} A - b^{2} - c^{2} = 0$, then

In $\Delta ABC$,$({b^2} - {c^2})\cot A + ({c^2} - {a^2})\cot B + ({a^2} - {b^2})\cot C = $

In triangle $ABC$,if $\angle A = 45^\circ$ and $\angle B = 75^\circ$,then $a + c\sqrt{2} = $

In $\triangle ABC$,find the value of $\sqrt{\frac{r r_2}{r_3 r_1}}$.

Let $ABC$ and $ABC^{\prime}$ be two non-congruent triangles with sides $AB=4$,$AC=AC^{\prime}=2\sqrt{2}$ and $\angle B=30^{\circ}$. The absolute value of the difference between the areas of these triangles is

Vedclass Test Series

Exam Paper Generator

Online Exam Module