To find the centroid of the triangle formed by the points A(cos θ, sin θ), B(sin θ, -cos θ), and the origin O(0, 0), and to determine the circle on which this centroid lies, follow these steps:
### Step 1: Identify the Coordinates of the Points
The coordinates of the points are:
- A = (cos θ, sin θ)
- B = (sin θ, -cos θ)
- O = (0, 0)
### Step 2: Calculate the Centroid of the Triangle
The centroid (G) of a triangle formed by points (x1, y1), (x2, y2), and (x3, y3) is given by the formula:
\[
G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)
\]
Substituting the coordinates of points A, B, and O:
\[
G\left(\frac{\cos \theta + \sin \theta + 0}{3}, \frac{\sin \theta - \cos \theta + 0}{3}\right)
\]
This simplifies to:
\[
G\left(\frac{\cos \theta + \sin \theta}{3}, \frac{\sin \theta - \cos \theta}{3}\right)
\]
### Step 3: Set the Coordinates of the Centroid
Let:
- \( x = \frac{\cos \theta + \sin \theta}{3} \)
- \( y = \frac{\sin \theta - \cos \theta}{3} \)
### Step 4: Express x and y in Terms of θ
From the equations:
1. \( 3x = \cos \theta + \sin \theta \)
2. \( 3y = \sin \theta - \cos \theta \)
### Step 5: Square and Add the Equations
To find a relationship between x and y, we square both equations and add them:
\[
(3x)^2 + (3y)^2 = (\cos \theta + \sin \theta)^2 + (\sin \theta - \cos \theta)^2
\]
Calculating the right side:
\[
(\cos^2 \theta + 2\sin \theta \cos \theta + \sin^2 \theta) + (\sin^2 \theta - 2\sin \theta \cos \theta + \cos^2 \theta)
\]
This simplifies to:
\[
2(\cos^2 \theta + \sin^2 \theta) = 2 \cdot 1 = 2
\]
Thus, we have:
\[
9x^2 + 9y^2 = 2
\]
### Step 6: Simplify the Equation
Dividing the entire equation by 9:
\[
x^2 + y^2 = \frac{2}{9}
\]
### Step 7: Identify the Circle's Center and Radius
The equation \( x^2 + y^2 = r^2 \) represents a circle centered at the origin (0, 0) with radius \( r \).
From our equation:
- Center: (0, 0)
- Radius: \( r = \sqrt{\frac{2}{9}} = \frac{\sqrt{2}}{3} \)
### Final Answer
The center of the circle is (0, 0) and the radius is \( \frac{\sqrt{2}}{3} \).
---
