Triangle with One Point at (0,0)

What is a Triangle with One Vertex at $(0, 0)$?

In coordinate geometry, a triangle with one vertex at the origin $(0, 0)$ is a common configuration for problems involving geometry and coordinate-based calculations.

The other two vertices of the triangle, say $(x_1, y_1)$ and $(x_2, y_2)$, are located elsewhere in the Cartesian plane.

This setup simplifies calculations for:

1. Area of the triangle.
2. Length of the sides.
3. Slopes and equations of the sides.

Area of the Triangle

The formula for the area of a triangle with vertices at $(0, 0)$, $(x_1, y_1)$, and $(x_2, y_2)$ is:

$$\text{Area} = \frac{1}{2} \left| x_1y_2 - x_2y_1 \right|$$

Length of the Sides

1. From $(0, 0)$ to $(x_1, y_1)$:

$$\text{Length} = \sqrt{x_1^2 + y_1^2}$$

2. From $(0, 0)$ to $(x_2, y_2)$:

$$\text{Length} = \sqrt{x_2^2 + y_2^2}$$

3. From $(x_1, y_1)$ to $(x_2, y_2)$:

$$\text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Equations of the Sides

1. From $(0, 0)$ to ($x_1, y_1$)****: Slope $m = \frac{y_1}{x_1}$, equation $y = mx$

2. From $(0, 0)$ to $(x_2, y_2)$: Slope $m = \frac{y_2}{x_2}$, equation $y = mx$

3. From $(x_1, y_1)$ to $(x_2, y_2)$: Slope $m = \frac{y_2 - y_1}{x_2 - x_1}$, equation: $y - y_1 = m(x - x_1)$

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Worked Examples

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Summary

• A triangle with one vertex at $(0, 0)$ simplifies calculations in coordinate geometry.
• Key formulas:
  • Area: $\frac{1}{2} \left| x_1y_2 - x_2y_1 \right|$
  • Side lengths: Use the distance formula for each pair of points.
  • Sides' equations: Use the slope and point-slope formulas.
• Practice applying these techniques to master solving geometric problems.