Key Concepts and Formulas

• Incenter of a Triangle: The incenter is the intersection of the angle bisectors of a triangle. Its coordinates are given by $I = \left( \frac{ax_1 + bx_2 + cx_3}{a+b+c}, \frac{ay_1 + by_2 + cy_3}{a+b+c} \right)$, where $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are the vertices, and $a$, $b$, and $c$ are the lengths of the sides opposite those vertices, respectively.
• Inradius of a Right Triangle: For a right-angled triangle with legs $a$ and $b$ and hypotenuse $c$, the inradius $r$ is given by $r = \frac{a+b-c}{2}$. The incenter of a right triangle with vertices at the origin and intercepts on the positive x and y axes is located at $(r, r)$.
• Intercepts of a Line: To find the x-intercept of a line, set $y=0$ and solve for $x$. To find the y-intercept, set $x=0$ and solve for $y$.

Step-by-Step Solution

Step 1: Find the x-intercept (Point A)

To find where the line $3x + 4y - 24 = 0$ intersects the x-axis, we set $y = 0$ and solve for $x$: $3x + 4(0) - 24 = 0$ $3x = 24$ $x = 8$ So, point A is $(8, 0)$.

Step 2: Find the y-intercept (Point B)

To find where the line $3x + 4y - 24 = 0$ intersects the y-axis, we set $x = 0$ and solve for $y$: $3(0) + 4y - 24 = 0$ $4y = 24$ $y = 6$ So, point B is $(0, 6)$.

Step 3: Determine the side lengths of triangle OAB

We have the vertices O(0, 0), A(8, 0), and B(0, 6). This is a right-angled triangle.

• $OA = a = 8$ (length of the side opposite to vertex B)
• $OB = b = 6$ (length of the side opposite to vertex A)
• $AB = c = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$ (length of the side opposite to vertex O)

Step 4: Calculate the inradius r

Since triangle OAB is a right triangle, we can use the formula for the inradius: $r = \frac{a + b - c}{2} = \frac{8 + 6 - 10}{2} = \frac{4}{2} = 2$

Step 5: Determine the coordinates of the incenter

Since the triangle is a right triangle with vertices at the origin and on the positive x and y axes, the incenter is located at $(r, r)$. Therefore, the incenter is $(2, 2)$.

Alternative Method (Using the General Formula for the Incenter)

We can use the formula $I = \left( \frac{ax_1 + bx_2 + cx_3}{a+b+c}, \frac{ay_1 + by_2 + cy_3}{a+b+c} \right)$, where $A(8,0)$, $B(0,6)$, $O(0,0)$, $a=6$, $b=8$, $c=10$: $I = \left( \frac{6(8) + 8(0) + 10(0)}{6+8+10}, \frac{6(0) + 8(6) + 10(0)}{6+8+10} \right)$ $I = \left( \frac{48}{24}, \frac{48}{24} \right) = (2, 2)$

Common Mistakes & Tips

• Confusing Incenter and Centroid/Circumcenter: Remember that the incenter is the intersection of angle bisectors, not medians (centroid) or perpendicular bisectors (circumcenter).
• Incorrectly Calculating Side Lengths: Double-check your calculations of the side lengths, especially when using the distance formula.
• Forgetting the Inradius Formula: The formula $r = \frac{a+b-c}{2}$ is only applicable to right triangles.

Summary

We found the x and y intercepts of the given line to determine the vertices of the right-angled triangle OAB. Then, we calculated the inradius of the triangle using the formula for right triangles and found the coordinates of the incenter to be (2, 2).

Final Answer

The final answer is $\boxed{(2, 2)}$, which corresponds to option (B).