the right triangle's C vertex is at the origin

In this scenario, you have a right triangle where vertex \( C \) is positioned at the origin, \((0, 0)\). Let me explain the implications of this setup:

1. **Coordinate System**: Since \( C \) is at the origin, the other two vertices, \( A(x_1, y_1) \) and \( B(x_2, y_2) \), will determine the orientation and dimensions of the triangle.
2. **Right Angle**: The triangle is a right triangle, meaning the right angle is at \( C \). Hence, the legs of the triangle must be along the coordinate axes. For example:
   - The line segment \( CA \) could be along the x-axis.
   - The line segment \( CB \) could be along the y-axis.
3. **Lengths of Sides**: The lengths of the legs of the triangle would correspond to the x and y coordinates of points \( A \) and \( B \). The hypotenuse will be the line segment \( AB \), and its length can be calculated using the Pythagorean theorem.

### Example:
If vertex \( A \) is at \( (a, 0) \) and vertex \( B \) is at \( (0, b) \), then:
- The length of leg \( CA \) is \( |a| \).
- The length of leg \( CB \) is \( |b| \).
- The hypotenuse \( AB \) has length \( \sqrt{a^2 + b^2} \) (using the Pythagorean theorem).

Would you like to explore this with a specific set of coordinates or perhaps solve for missing dimensions?

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### Related Questions:
1. How do you find the area of the right triangle when the coordinates of all vertices are known?
2. How can you find the equation of the hypotenuse in this scenario?
3. What is the relationship between the slopes of the legs in a coordinate-based right triangle?
4. How can we calculate the distance between two points in this context using the distance formula?
5. Can you use trigonometry to find the angles of the right triangle given the lengths of the legs?

#### Tip:
The slopes of the legs of a right triangle are perpendicular, so their product is always \( -1 \).

Explore how to work with a right triangle where vertex C is at the origin (0,0) in coordinate geometry. This problem uses key geometric concepts such as the Pythagorean theorem and distance formula to calculate side lengths and explore triangle properties. Suitable for Grades 8-10.

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