Finding Point P Coordinates Equidistant from Vertices

The point P that is equidistant from the three vertices of a triangle is known as the circumcenter. Let the coordinates of P be $(x, y)$.

Triangle Vertex Coordinates

The vertices of the triangle are given as:

• A: $(0, 0)$
• B: $(4, 0)$
• C: $(0, 3)$

Identifying Triangle Type

We can observe the coordinates. Vertex A is at the origin $(0, 0)$. Vertex B lies on the x-axis, and vertex C lies on the y-axis. This forms a right-angled triangle with the right angle at vertex A.

Calculating Circumcenter for Right Triangle

For a right-angled triangle, the circumcenter is located at the midpoint of the hypotenuse.

The hypotenuse connects the vertices B$(4, 0)$ and C$(0, 3)$.

The midpoint formula is given by $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Applying this formula to find the midpoint of BC:

x-coordinate of P $= \frac{4 + 0}{2} = \frac{4}{2} = 2$

y-coordinate of P $= \frac{0 + 3}{2} = \frac{3}{2} = 1.5$

Resulting Coordinates

Therefore, the coordinates of point P are $(2, 1.5)$.