Tangent to a Circle
Understand the core property of a **tangent** line: that the radius drawn to the point of tangency is always **perpendicular** to the tangent line.
Help & Instructions
â–¼- **Adjust Slope:** Use the slider to change the slope of the line passing through the point P.
- **Find Tangent:** Try to visually align the line so it touches the circle at exactly point P.
- **Check Angle:** The calculated Angle is the angle between the line and the radius. Your goal is to make this angle as close to $90^\circ$ as possible.
- **Solve:** Click "Verify Perpendicularity" when you think your line is a perfect tangent!
- Verify the theorem: Radius is perpendicular to the tangent at the point of contact.
- Understand the relationship between the slope of the radius and the slope of the tangent.
- Reinforce geometric concepts through visual adjustment.
The Tangency Target 🎯
A **tangent** to a circle is a line that intersects the circle at exactly one point, the **point of tangency**. The most important property is that the radius drawn to this point of tangency is always **perpendicular** ($90^\circ$) to the tangent line.
The Mathematics Behind the Puzzles
If two lines are perpendicular, the product of their slopes is $-1$. If the slope of the radius ($m_r$) is $\frac{y_p - y_c}{x_p - x_c}$, then the slope of the tangent ($m_t$) must satisfy:
$$m_t \times m_r = -1$$This means $m_t = -1/m_r$. This geometric principle is used to define the correct tangent line.
The concept of tangency is used in:
- **Engineering:** Designing smooth curves for roads and roller coasters.
- **Optics:** Analyzing the path of light rays that graze curved surfaces.
- **Astronomy:** Calculating the trajectory of planets or satellites.
