Sure, let's solve the problem!
We are given the endpoints of the hypotenuse of a right-angle triangle, which are [tex]\((0, 6)\)[/tex] and [tex]\((6, 0)\)[/tex]. We need to determine the equation of the locus of the third vertex, [tex]\((x, y)\)[/tex], of the right-angle triangle.
The key idea here is that the third vertex, [tex]\((x, y)\)[/tex], is equidistant from both endpoints of the hypotenuse [tex]\((0, 6)\)[/tex] and [tex]\((6, 0)\)[/tex]. This is because the third vertex is at the right angle to the hypotenuse, forming two congruent right triangles.
We will use the property that for any point [tex]\((x, y)\)[/tex] on the locus, the distances to the points [tex]\((0, 6)\)[/tex] and [tex]\((6, 0)\)[/tex] must be equal. This allows us to set up the equation as follows:
### Step-by-step Solution:
1. Express the distances using the distance formula:
The distance between [tex]\((x, y)\)[/tex] and [tex]\((0, 6)\)[/tex] is:
[tex]\[
\text{Distance}_1 = \sqrt{(x - 0)^2 + (y - 6)^2}
\][/tex]
The distance between [tex]\((x, y)\)[/tex] and [tex]\((6, 0)\)[/tex] is:
[tex]\[
\text{Distance}_2 = \sqrt{(x - 6)^2 + (y - 0)^2}
\][/tex]
2. Set up the equation stating these distances are equal:
[tex]\[
\sqrt{(x - 0)^2 + (y - 6)^2} = \sqrt{(x - 6)^2 + (y - 0)^2}
\][/tex]
3. Square both sides to eliminate the square roots:
[tex]\[
(x - 0)^2 + (y - 6)^2 = (x - 6)^2 + (y - 0)^2
\][/tex]
4. Simplify the equation:
[tex]\[
x^2 + (y - 6)^2 = (x - 6)^2 + y^2
\][/tex]
5. Distribute and expand both sides:
[tex]\[
x^2 + y^2 - 12y + 36 = x^2 - 12x + 36 + y^2
\][/tex]
6. Cancel out common terms on both sides:
Subtract [tex]\(x^2 + y^2 + 36\)[/tex] from both sides:
[tex]\[
-12y = -12x
\][/tex]
7. Solve for [tex]\(y\)[/tex]:
Dividing both sides of the equation by [tex]\(-12\)[/tex], we get:
[tex]\[
y = x
\][/tex]
### Conclusion:
The equation of the locus of the third vertex of the right-angle triangle is:
[tex]\[
y = x
\][/tex]
Thus, the third vertex must lie on the line [tex]\(y = x\)[/tex] to maintain the right-angle property with the given hypotenuse endpoints [tex]\((0, 6)\)[/tex] and [tex]\((6, 0)\)[/tex].
