Question

Write the coordinates of the in-centre of the triangle with vertices at (0, 0), (5, 0) and (0, 12).

Answer 1

Let A(0,0), B(5, 0) and C(0, 12) be the vertices of the given triangle.
In-centre I of a triangle with vertices

\[A\left( x_1 , y_1 \right), B\left( x_2 , y_2 \right)\text{ and }C\left( x_3 , y_3 \right)\]

\[\text{ I }\equiv \left( \frac{a x_1 + b x_2 + c x_3}{a + b + c}, \frac{a y_1 + b y_2 + c y_3}{a + b + c} \right)\]
Now,
\[a = BC = \sqrt{\left( 5 - 0 \right)^2 + \left( 0 - 12 \right)^2} = 13\]
\[b = AC = \sqrt{0 + {12}^2} = 12\]
\[c = AB = \sqrt{0 + 5^2} = 5\]

\[\therefore\text{ I }\equiv \left( \frac{13 \times 0 + 12 \times 5 + 5 \times 0}{13 + 12 + 5}, \frac{13 \times 0 + 12 \times 0 + 5 \times 12}{13 + 12 + 5} \right)\]
\[ \Rightarrow\text{ I }\equiv \left( \frac{60}{30}, \frac{60}{30} \right) = \left( 2, 2 \right)\]

Hence, the coordinates of the in-centre of the triangle with vertices at (0, 0), (5, 0) and (0, 12) is (2, 2).