Tangent to a Circle: Definition & Properties

Tangent to a Circle

Welcome to the explanation of tangents to a circle! This is a fundamental concept in geometry, essential for understanding shapes and their properties. This guide breaks down the concept in a clear and concise manner, perfect for high school students.

A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of contact. The tangent line lies in the same plane as the circle.

Think of it like a straight road just grazing the edge of a circular park. The point where the road touches the park’s boundary is the point of contact.

Formulae

While there isn’t a single formula for “tangents” in the same way there are formulas for area or volume, understanding the relationship between tangents, radii, and the circle’s center is key. Key concepts include:

The tangent is perpendicular to the radius at the point of contact. This is the most crucial relationship.
Using the Pythagorean theorem ($a^2 + b^2 = c^2$) can often be used in problems involving tangents, radii, and secants.

Examples

Example-1:

A circle has a radius of 5 cm. A tangent is drawn from an external point P to the circle. The distance from P to the center of the circle is 13 cm. Find the length of the tangent.
Solution: Let the center of the circle be O, and the point of contact of the tangent be T. Then, OT is the radius (5 cm), OP is the distance from the external point to the center (13 cm), and PT is the tangent we want to find. Since the tangent is perpendicular to the radius at the point of contact, triangle OTP is a right-angled triangle. Using the Pythagorean theorem: $OT^2 + PT^2 = OP^2$ $5^2 + PT^2 = 13^2$ $25 + PT^2 = 169$ $PT^2 = 144$ $PT = \sqrt{144} = 12$ cm Therefore, the length of the tangent is 12 cm.

Example-2:

Two tangents, PA and PB, are drawn to a circle from an external point P. If the angle between the tangents (angle APB) is 60 degrees, and the radius of the circle is 6 cm, find the length of the chord AB.
Solution: Since tangents from an external point to a circle are equal in length, PA = PB. Therefore, triangle PAB is an isosceles triangle. Given that angle APB is 60 degrees, and since the sum of the angles in a triangle is 180 degrees, and PA=PB therefore angle PAB = angle PBA. Hence, triangle PAB is an equilateral triangle (all angles are 60 degrees). So, PA = PB = AB. Let O be the center, and let OA = OB = 6cm (radii). The line segment from the center to the external point, PO, bisects angle APB and the chord AB, meaning PO is the perpendicular bisector. Thus, since PAB is equilateral, PA = AB = 6 cm. Therefore, the length of the chord AB is also equal to 6 cm.

Theorem with Proof

Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Proof:

Given: A circle with center O, and a tangent AB at a point P on the circle. OP is the radius.
To Prove: OP is perpendicular to AB (i.e., $\angle OPA = 90^\circ$).
Construction: Take a point Q, other than P, on the tangent AB. Join O to Q.
Proof:
- Since Q lies on the tangent AB, and Q is not P, Q must lie outside the circle (because a tangent touches the circle at only one point).
- Therefore, OQ is longer than OP (OQ > OP), as any other point on the tangent other than the contact point will be farther from the center.
- OP is the shortest distance from O to the line AB (because it’s a radius to the point of contact).
- Since the shortest distance from a point to a line is the perpendicular distance, OP must be perpendicular to AB.
- Therefore, $\angle OPA = 90^\circ$.
Hence proved.

Common mistakes by students

Assuming tangents are always parallel: Students often incorrectly assume tangents are always parallel, especially when drawing diagrams. Tangents are parallel only if drawn at opposite ends of a diameter.
Forgetting the Perpendicular Relationship: The most common mistake is forgetting that a tangent is perpendicular to the radius at the point of contact. This is crucial for solving problems.
Confusing Secants and Tangents: Students sometimes mix up secants (lines intersecting a circle at two points) with tangents.

Real Life Application

Engineering: Tangents are essential in designing curves in roads, railways, and roller coasters, ensuring smooth transitions.
Architecture: Architects use tangent principles to create curved structures and ensure structural integrity.
Computer Graphics: Tangents are used extensively in creating smooth curves and surfaces in computer-generated images and animations.

Fun Fact

The concept of tangents and circles has been studied since ancient times, even dating back to the work of Euclid! Understanding tangents is also crucial for understanding more advanced topics like calculus, where derivatives are used to find the slope of a tangent line to a curve.