Question

If \( O(0, 0, 0), A(3, 0, 0), B(0, 4, 0) \) form a triangle then the incenter of triangle OAB is:

• A. \( (0, 1, 0) \)
• B. \( (0, 1, 1) \)
• C. \( (1, 0, 1) \)
• D. \( (1, 1, 0) \)

Hint:

The incenter of a triangle can be calculated using the side lengths as weights for the vertex coordinates.

Answer 1

The Correct Option is D

Step 1: Calculate the distances of the sides OA, AB, and BO, which are 3, 5, and 4 respectively. 
Step 2: Use the formula for incenter coordinates \( Ix = \frac{a \cdot x_1 + b \cdot x_2 + c \cdot x_3}{a+b+c} \), \( Iy = \frac{a \cdot y_1 + b \cdot y_2 + c \cdot y_3}{a+b+c} \), and \( Iz = \frac{a \cdot z_1 + b \cdot z_2 + c \cdot z_3}{a+b+c} \), where \( a, b, c \) are the lengths opposite to the vertices \( A, B, C \) respectively. 
Step 3: Plug in the values and calculate: \[ Ix = \frac{3 \cdot 0 + 5 \cdot 0 + 4 \cdot 3}{3+5+4} = 1, \] \[ Iy = \frac{3 \cdot 0 + 5 \cdot 4 + 4 \cdot 0}{3+5+4} = 1, \] \[ Iz = 0. \]