To find the circumcenter of the triangle with vertices at (0,0), (4a,0), and (0,6a), we can follow these steps:
### Step 1: Identify the vertices of the triangle
The vertices of the triangle are given as:
- A(0, 0)
- B(4a, 0)
- C(0, 6a)
### Step 2: Determine the hypotenuse
Since the triangle formed by these points is a right triangle (with the right angle at A), the hypotenuse will be the line segment connecting points B and C.
### Step 3: Find the midpoint of the hypotenuse
To find the circumcenter of a right triangle, we need to calculate the midpoint of the hypotenuse (line segment BC). The midpoint formula is given by:
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
For points B(4a, 0) and C(0, 6a), we can substitute the coordinates into the midpoint formula.
### Step 4: Substitute the coordinates into the midpoint formula
Let:
- \(x_1 = 4a\), \(y_1 = 0\) (coordinates of B)
- \(x_2 = 0\), \(y_2 = 6a\) (coordinates of C)
Now, substituting into the formula:
\[
\text{Midpoint} = \left( \frac{4a + 0}{2}, \frac{0 + 6a}{2} \right)
\]
### Step 5: Simplify the expressions
Calculating the x-coordinate:
\[
\frac{4a + 0}{2} = \frac{4a}{2} = 2a
\]
Calculating the y-coordinate:
\[
\frac{0 + 6a}{2} = \frac{6a}{2} = 3a
\]
### Step 6: Write the coordinates of the circumcenter
Thus, the coordinates of the circumcenter (midpoint of hypotenuse BC) are:
\[
\text{Circumcenter} = (2a, 3a)
\]
### Final Answer:
The circumcenter of the triangle is at the point \((2a, 3a)\).
---
