What is the 90-degree angle circle theorem?

90-Degree Angle Circle Theorem

In geometry, the 90-degree angle circle theorem states that when a line is tangent to a circle, it forms a 90-degree angle with the radius of the circle at the point of tangency.

Proof:

Let's consider a circle with center O and radius r. Let P be a point outside the circle, and let PQ be a tangent to the circle at point Q. Draw radius OQ from the center to the point of tangency.

Since PQ is a tangent, it is perpendicular to OQ at point Q. Therefore, angle PQQ is 90 degrees.

Consequences:

• Equal Tangents from External Point: When two tangents are drawn from an external point to a circle, they are equal in length.

To prove this, consider two tangents, PT and PS, drawn from point P to the circle. Join P to the center O. Since both PT and PS are tangents, angles PTQ and PSQ are both 90 degrees. Therefore, triangles PQT and PQS are both right triangles.

By the Pythagorean Theorem, PT² = PQ² - QT² and PS² = PQ² - QS². Since PQ is common to both triangles, we have PT² = PS². Therefore, PT = PS.

Applications:

The 90-degree angle circle theorem has many applications in geometry, including:

• Determining the length of a tangent
• Finding the equation of a tangent
• Constructing circles and arcs
• Solving geometry problems involving circles and tangents