The product of the arithmetic mean of the len | vedclass

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

Question

(B) Let the sides of the triangle be $a, b, c$ and the corresponding altitudes be $h_1, h_2, h_3$.We know that $a h_1 = b h_2 = c h_3 = 2 \Delta$.Thus

Vedclass · Accepted Answer

$2 \Delta$

Vedclass · Answer

(B) Let the sides of the triangle be $a, b, c$ and the corresponding altitudes be $h_1, h_2, h_3$.We know that $a h_1 = b h_2 = c h_3 = 2 \Delta$.Thus,$h_1 = \frac{2 \Delta}{a}$,$h_2 = \frac{2 \Delta}{b}$,and $h_3 = \frac{2 \Delta}{c}$.The arithmetic mean of the sides is $AM = \frac{a + b + c}{3}$.The harmonic mean of the altitudes is $HM = \frac{3}{\frac{1}{h_1} + \frac{1}{h_2} + \frac{1}{h_3}}$.Substituting the values of $h_1, h_2, h_3$:$\frac{1}{h_1} + \frac{1}{h_2} + \frac{1}{h_3} = \frac{a}{2 \Delta} + \frac{b}{2 \Delta} + \frac{c}{2 \Delta} = \frac{a + b + c}{2 \Delta}$.Therefore,$HM = \frac{3}{\frac{a + b + c}{2 \Delta}} = \frac{6 \Delta}{a + b + c}$.The product $AM 	imes HM = \left(\frac{a + b + c}{3}ight) 	imes \left(\frac{6 \Delta}{a + b + c}ight) = 2 \Delta$.

The product of the arithmetic mean of the lengths of the sides of a triangle and the harmonic mean of the lengths of the altitudes of the triangle is equal to: [where $\Delta$ is the area of the triangle $ABC$]

Similar Questions

If $\alpha$ and $\beta$ are solutions of $\sin^2 x + a \sin x + b = 0$ as well as $\cos^2 x + c \cos x + d = 0$,then $\sin(\alpha + \beta)$ is equal to

In a $\Delta ABC$,$a = a_1 = 2$,$b = a_2$,$c = a_3$ such that $a_{p+1} = \frac{5^p}{3^{2-p}} a_p \left( 2^{2-p} - \frac{4p-2}{5^p} a_p \right)$ where $p = 1, 2$,then:

If $\Delta$ denotes the area of triangle $ABC$,then $(b \sin C + c \sin B)(b \cos C + c \cos B) =$

In a triangle $ABC$,if $\cos A \cos B + \sin A \sin B \sin C = 1$,then $a : b : c =$

If in $\Delta ABC,$ $a = 6, b = 3$ and $\cos(A - B) = \frac{4}{5},$ then its area will be ..... $square \, unit.$

Vedclass Test Series

Exam Paper Generator

Online Exam Module