Glossary
Angles in a Semicircle
An angle inscribed in a semicircle is always a right angle (90 degrees). This occurs when the inscribed angle intercepts a diameter.
Example:
Drawing a triangle inside a circle where one side is the diameter will always result in a 90-degree angle at the opposite vertex, thanks to angles in a semicircle.
Central Angle
An angle formed by two radii of a circle that meet at the circle's center. Its measure is equal to the measure of its intercepted arc.
Example:
If a slice of pizza forms a 60-degree central angle, the crust (arc) it cuts off is also 60 degrees.
Chord
A line segment connecting any two points on the circle's circumference. It does not necessarily pass through the center.
Example:
Imagine cutting a straight line across a pizza, not necessarily through the middle; that line is a chord.
Circumscribed Angle
An angle formed by two tangent lines that intersect outside the circle. Its measure is half the difference of the measures of the two intercepted arcs.
Example:
If you draw two lines from a single point that just touch a circle, the angle they form is a circumscribed angle.
Cyclic Quadrilateral
A quadrilateral inscribed in a circle, meaning all four of its vertices lie on the circle's circumference. Opposite angles of a cyclic quadrilateral are supplementary (add up to 180 degrees).
Example:
If you have a four-sided shape perfectly fitting inside a circle, it's a cyclic quadrilateral, and its opposite corners will always add up to 180 degrees.
Inscribed Angle
An angle formed by two chords that meet on the circle's circumference. Its measure is half the measure of its intercepted arc.
Example:
When you look at a specific arc from a point on the circle's edge, the inscribed angle you form will be half the size of the arc itself.
Inscribed Angle Theorem
This theorem states that inscribed angles that intercept the same arc are congruent (equal in measure).
Example:
If two different points on a circle's circumference both 'look at' the same inscribed angle theorem arc, the angles they form will be identical.
Intercepted Arc
The portion of the circle's circumference that lies between the two rays of an angle. Its measure is directly related to the angle that intercepts it.
Example:
A 90-degree intercepted arc means the central angle is 90 degrees, and any inscribed angle that intercepts it is 45 degrees.
Perpendicular Bisector of a Chord
A line that cuts a chord into two equal halves and is perpendicular to it. This line always passes through the circle's center.
Example:
To find the exact center of a circular table, you could draw a perpendicular bisector of a chord and know the center lies on that line.
Point of Tangency
The single point where a tangent line touches the circumference of a circle. The radius drawn to this point is perpendicular to the tangent line.
Example:
The exact spot where a bicycle tire meets the ground is its point of tangency.
Secant
A line that intersects a circle at two distinct points, creating a chord inside the circle. It extends infinitely in both directions.
Example:
A line that 'cuts through' a circle, entering on one side and exiting on the other, is a secant.
Secant-Tangent Theorem
For a secant and a tangent drawn from an external point to a circle, the square of the tangent's length is equal to the product of the whole secant's length and its external segment.
Example:
When one line just touches a circle and another cuts through it from the same external point, the secant-tangent theorem helps you find unknown lengths using the 'tangent squared equals whole times outside' formula.
Tangent
A line that touches a circle at exactly one point, known as the point of tangency. It is always perpendicular to the radius drawn to that point.
Example:
When a car tire just touches the road, the road acts as a tangent line to the circular tire.
Tangent Segments Theorem
States that two tangent segments drawn from the same external point to a circle are congruent (equal in length).
Example:
If you draw two lines from your hand to just touch a ball, the lengths of those lines from your hand to the ball will be equal due to the tangent segments theorem.
Tangent-Chord Angle
The angle formed between a tangent line and a chord at the point of tangency. Its measure is equal to the inscribed angle in the opposite segment.
Example:
When a line barely touches a circle and a chord starts from that touch point, the angle between them is a tangent-chord angle.
Two Secants Theorem
For two secants drawn from an external point to a circle, the product of the length of the whole secant and its external segment is equal for both secants.
Example:
If two lines cut through a circle from the same outside point, the 'whole times outside' rule applies to both, as per the two secants theorem.