What is the measure of an angle formed by a tangent of a circle and the radius drawn to the point of tangency?

A line that intersects a circle in exactly one point is called a tangent and the point where the intersection occurs is called the point of tangency. The tangent is always perpendicular to the radius drawn to the point of tangency.

A secant is a line that intersects a circle in exactly two points.

When a tangent and a secant, two secants, or two tangents intersect outside a circle then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

$$m\angle A=\frac{1}{2}(m\overline{DE}-m\overline{BC} )$$

When two chords intersect inside a circle, then the measures of the segments of each chord multiplied with each other is equal to the product from the other chord:

$$AB\cdot EB=CE\cdot ED$$

If two secants are drawn to a circle from one exterior point, then the product of the external segment and the total length of each secant are equal:

$$AB\cdot AD=AC\cdot AE$$

If one secant and one tangent are drawn to a circle from one exterior point, then the square of the length of the tangent is equal to the product of the external secant segment and the total length of the secant:

$$AB^{2}=AC\cdot AD$$

If we have a circle drawn in a coordinate plane, with the center in (a,b) and the radius r then we could always describe the circle with the following equation:

$$(x-a)^{2}+(y-b)^{2}=r^{2}$$

Video lesson

Find the value of t in the figure

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.

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.