To find the equation of the circle inscribed in the triangle with vertices A(1, 0), B(5, 0), C(3, \sqrt{5}}, we need to follow these steps:

Step 1: Calculate the sides of the triangle

Using the distance formula, the length of each side of the triangle is calculated as follows:

• $AB$ is the distance between $A(1, 0)$ and $B(5, 0)$: $AB = \sqrt{(5 - 1)^2 + (0 - 0)^2} = \sqrt{16} = 4$

• $AC$ is the distance between $A(1, 0)$ and C(3, \sqrt{5}}: $AC = \sqrt{(3 - 1)^2 + (\sqrt{5} - 0)^2} = \sqrt{4 + 5} = \sqrt{9} = 3$

• $BC$ is the distance between $B(5, 0)$ and C(3, \sqrt{5}}: $BC = \sqrt{(3 - 5)^2 + (\sqrt{5} - 0)^2} = \sqrt{(-2)^2 + (\sqrt{5})^2} = \sqrt{4 + 5} = \sqrt{9} = 3$

So, the sides of the triangle are: $AB = 4, \quad AC = 3, \quad BC = 3$

Step 2: Calculate the semi-perimeter $s$ of the triangle

The semi-perimeter $s$ is half the perimeter of the triangle: $s = \frac{AB + AC + BC}{2} = \frac{4 + 3 + 3}{2} = 5$

Step 3: Calculate the area of the triangle

The area $A$ of the triangle can be found using Heron's formula: $A = \sqrt{s(s - AB)(s - AC)(s - BC)} = \sqrt{5(5 - 4)(5 - 3)(5 - 3)} = \sqrt{5(1)(2)(2)} = \sqrt{20} = 2\sqrt{5}$

Step 4: Calculate the inradius $r$

The inradius $r$ of the inscribed circle is given by: $r = \frac{A}{s} = \frac{2\sqrt{5}}{5}$

Step 5: Find the coordinates of the incenter

The coordinates of the incenter $(x_{\text{incenter}}, y_{\text{incenter}})$ are given by the formula: $x_{\text{incenter}} = \frac{aA_x + bB_x + cC_x}{a + b + c}, \quad y_{\text{incenter}} = \frac{aA_y + bB_y + cC_y}{a + b + c}$ where $A_x, A_y$, $B_x, B_y$, and $C_x, C_y$ are the coordinates of the vertices, and $a$, $b$, and $c$ are the lengths of the sides opposite to those vertices.

For our triangle: $a = BC = 3, \quad b = AC = 3, \quad c = AB = 4$

Substitute the coordinates of $A(1, 0)$, $B(5, 0)$, and C(3, \sqrt{5}}: $x_{\text{incenter}} = \frac{3 \cdot 1 + 3 \cdot 5 + 4 \cdot 3}{3 + 3 + 4} = \frac{3 + 15 + 12}{10} = \frac{30}{10} = 3$ $y_{\text{incenter}} = \frac{3 \cdot 0 + 3 \cdot 0 + 4 \cdot \sqrt{5}}{3 + 3 + 4} = \frac{0 + 0 + 4\sqrt{5}}{10} = \frac{4\sqrt{5}}{10} = \frac{2\sqrt{5}}{5}$

So, the incenter is $\left( 3, \frac{2\sqrt{5}}{5} \right)$.

Step 6: Write the equation of the inscribed circle

The equation of a circle is given by: $(x - x_{\text{incenter}})^2 + (y - y_{\text{incenter}})^2 = r^2$ Substitute $x_{\text{incenter}} = 3$, $y_{\text{incenter}} = \frac{2\sqrt{5}}{5}$, and $r = \frac{2\sqrt{5}}{5}$: $(x - 3)^2 + \left( y - \frac{2\sqrt{5}}{5} \right)^2 = \left( \frac{2\sqrt{5}}{5} \right)^2$ $(x - 3)^2 + \left( y - \frac{2\sqrt{5}}{5} \right)^2 = \frac{20}{25} = \frac{4}{5}$

Thus, the equation of the inscribed circle is: $(x - 3)^2 + \left( y - \frac{2\sqrt{5}}{5} \right)^2 = \frac{4}{5}$

Would you like more details on any of these steps?

Related Questions:

1. How do you calculate the centroid of a triangle?
2. What is the significance of the incenter of a triangle?
3. How does Heron’s formula help in calculating the area of any triangle?
4. What are other methods to find the equation of an inscribed circle in a triangle?
5. How can we extend this method to find the circumcircle of the same triangle?

Tip:

Always verify your distance and area calculations by cross-checking with alternate methods like the coordinate geometry or trigonometry approaches.