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HotmathMath Homework. Do It Faster, Learn It Better.

Tangent to a Circle

A tangent to a circle is a straight line which touches the circle at only one point. This point is called the point of tangency.

The tangent to a circle is perpendicular to the radius at the point of tangency.

Math diagram

In the circle O , P T is a tangent and O P ¯ is the radius.

If P T is a tangent, then O P ¯ is perpendicular to P T .

For example, suppose O P ¯ = 3 units and P T ¯ = 4 units. Find the length of O T ¯ .

Because the radius is perpendicular to the tangent at the point of tangency, O P ¯ P T .

This makes the angle P a right angle in the triangle O P T and triangle O P T a right triangle.

Now use the Pythagorean Theorem to find O T ¯ .

( O P ) 2 + ( P T ) 2 = ( O T ) 2 3 2 + 4 2 = ( O T ) 2 9 + 16 = ( O T ) 2 25 = ( O T ) 2 ± 5 = O T

Since the length cannot be negative, the length of O T ¯ is 5 units.

 

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