Understanding Tangents for 14-Year-Old Students: Basics and How to Find Them

What is a Tangent?

In geometry, a tangent is a straight line that touches a curve at exactly one point without crossing it. The most common example is a tangent to a circle.

Tangent to a Circle

For a circle, a tangent line touches the circle at exactly one point, called the point of tangency. At this point, the line does not cross into the inside of the circle but just 'kisses' the circle.

Key Properties:

• The tangent line is perpendicular (at a right angle) to the radius of the circle drawn to the point of tangency.
• They meet exactly at one point.

How to Find a Tangent to a Circle

Imagine you have a circle with a center at point O and a radius called r. Here's a simple way to think about tangent lines:

1. Find the point of tangency: This is where the tangent line touches the circle.
2. Draw the radius from the center O to this point: The tangent line will be perpendicular (at 90 degrees) to this radius.
3. Use this perpendicularity to find the equation of the tangent line: If you know the circle's equation and the point, you can use algebra to find the line's equation.

Example:

Consider a circle centered at the origin (0,0) with radius 5, described by the equation x² + y² = 25. If you pick the point (3,4) on the circle, the radius is the line from (0,0) to (3,4).

The slope of this radius is 4/3. The tangent line at (3,4) will have a slope that is the negative reciprocal, which is -3/4 (because the tangent is perpendicular to the radius).

Using the point-slope form of a line:

y - 4 = -³/₄(x - 3)

This is the equation of the tangent line to the circle at point (3,4).