Finding centroid's coordinates using Pappus theorem

by Nicko   Last Updated September 11, 2019 09:20 AM - source

enter image description here

The task is to find the centroid of the given triangle (see the image above). We also should use the fact that the volume of a cone of radius r and height h is $V = \frac{1}{3}\Pi r^2h$. My solution:
1) Denote the dots as following: O(0,0), R(a, 0), Q(a, b), P(a, c);
2) Find the volume of the cone with radius OR and height PR:

$V = \frac{1}{3}\Pi r^2h$ = $\frac{1}{3}\Pi a^2c$

3) Find the volume of the cone with radius OR and height RQ:

$V = \frac{1}{3}\Pi r^2h$ = $\frac{1}{3}\Pi a^2b$

4) If we rotate triangle OQP about the line x = a, then according to Pappus theorem it's volume is equal to:

$V = 2\Pi pA$, where

p - distance from centroid of OQP to axis of revolution (x = a);
A - area of triangle OQP;

5) The next step is to find area of OQP. We will do this with formula for right triangle $\frac{1}{2}ab$ (a and b are edges). The area of triangle ORP is equal to $\frac{1}{2} ac$, and the area of OQR is equal to $\frac{1}{2} ab$. Thus, the area of OQP is equal:

$A = \frac{1}{2} ac - \frac{1}{2} ab = \frac{1}{2} a(c-b)$

6) We can now find the volume of solid, which will be created by revolving triangle OQP about line x = a using the following approaches:

$V = \frac{1}{3}\Pi a^2c - \frac{1}{3}\Pi a^2b$ (difference of cone volumes)

$V = 2\Pi pA = 2\Pi p\frac{1}{2} a(c-b)$ (Pappus Theorem)

This to expressions are equal. We simplify the equality and receive that $p = \frac{a}{3}$ (the distance from centroid to axis of revolution (x = a)).

7) As this distance is equal to $p = \frac{a}{3}$, then x coordinate of centroid is equal to $p = \frac{2a}{3}$. Also, we know that centroid of triangle is on median. So we draw the line between dots (0, 0) and (a, $\frac{c+b}{2}$). This line has an equation $y = \frac{c+b}{2a}x$. If we replace x with $\frac{2a}{3}$, the y is equal $\frac{c+b}{3}$. So, the coordinates of centroid is ($\frac{2a}{3}$, $\frac{c+b}{3}$).

However, the textbook I use give the following answer ($\frac{2a(a-b}{3(c-b)}$, $\frac{c+b}{3}$).

I will be very grateful if anybody can explain why the answers is different. Thank you in advance! (Please, note that y coordinate is correct.)



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