Properties of Circles: Tangents and Perpendicularity

This video tutorial explores the properties of circles, focusing on tangents and their unique characteristics. It begins by differentiating between secant and tangent lines, then delves into a theorem stating that a tangent is perpendicular to the radius at the point of tangency. The theorem is proven using contradiction. The video also covers the tangent segments theorem, which states that tangent segments from an external point are congruent. An example problem is solved to illustrate these concepts.

What is the main difference between a tangent line and a secant line in a circle?

• A tangent line intersects the circle at only one point.

• A tangent line intersects the circle at two points.

• A tangent line is parallel to the radius.

• A tangent line is longer than a secant line.

According to the proof by contradiction, what is assumed about the line OP in relation to the tangent line M?

• OP is parallel to M.

• OP is not perpendicular to M.

• OP is tangent to M.

• OP is equal in length to M.

What geometric principle is used to prove that a tangent is perpendicular to the radius at the point of tangency?

• The angle sum property of triangles

• The shortest distance between a point and a line

• The concept of parallel lines

• The Pythagorean Theorem

What does the theorem about tangent segments from an external point state?

• Tangent segments are equal to the diameter.

• Tangent segments are always longer than the radius.

• Tangent segments from an external point are congruent.

• Tangent segments are perpendicular to each other.

In the example problem, if SK is x + 4 and SM is 2x - 9, what is the value of x?

• x = 5

• x = 13

• x = 9

• x = 4