A tangent line, or simply a "tangent", is a straight line that touches a curve at exactly one point. In higher mathematics, the equation of the tangent line is useful because it tells you the slope of the curve at a specific instance of a point on that curve. In this lesson, we will focus on tangents to a circle.
Properties of Tangents to a Circle
1) A tangent line touches the circle at exactly 1 point.
The point that the tangent meets the circle is called the point of tangency. You can think of this point as the point where the tangent is infinitely close to the curve's edge. Lines with two points of intersection with curves are called secants.
2) The tangent line is perpendicular to the radius of the circle at the point of tangency.
The radius of the circle from the center to the point of tangency forms a right angle with the tangent line. This property is known as the Radius Tangent Theorem, and is especially useful in angle calculations for triangles and quadrilaterals .
3) Two tangents form equal angles with the chord that connects their points of tangencies.
In addition, the lengths of the two tangents from their intersecting point to their respective points of tangencies are equal. This property is called the Two Tangent Theorem.
Example 1. Line AC is tangent to circle O at point C. Find <OAC and <OBC.
The radius tangent theorem tells us that <OCA = 90 degrees. This is because radius OC is connected to the point of tangency of AC.
Because <BOC = 63 and <OCA = 90, we can use the fact that the interior angles of a triangle add up to 180 degrees to figure out <OAC.
We have <OAC = 180 - 90 - 63 = 27 degrees.
If we draw an imaginary line connecting points BC, we will form an isosceles triangle OBC with a vertex at O. Based on what we know about isosceles triangles, <OBC = <OCB. Therefore, <OBC = (180 - 63)/2 = 58.5 degrees.
Example 2. Find the perimeter of quadrilateral ABCD.
From the Two Tangent Theorem, BT = BS = 3.3 units. Thus, BC = BS + SC = 3.3 + 5.1 = 8.4 units.
From the Two Tangent Theorem, CS = CR = 5.1 units. Thus, DR = DC = CR = 12.1 - 5.1 = 7.0 units.
From the Two Tangent Theorem, DR = DQ = 7.0 units, so AQ = AD - DQ = 12.2 - 7.0 = 5.2 units. Since AT = AQ = 5.2 units, AB = AT + TB = 5.2 + 3.3 = 8.5 units.
The perimeter is found by adding up all four side lengths. We have Perimeter = AB + BC + CD + DA = 8.5 + 8.4 + 12.1 + 12.2 = 41.2 units.