Circle

Central angle. Arc of a circle.☰

A circle is a set of points in a plane that are equidistant from a fixed point called the center. The circle is an important shape in mathematics, and it is used in many fields, including geometry, trigonometry, and calculus.
Central angle is an angle whose vertex is the center of the circle. It is formed by two radii of the circle that connect the center to two points on the circle. In other words, a central angle is an angle whose vertex is at the center of a circle and whose arms intersect two points on the circle.
Consider a circle with center O, and let A and B be two points on the circle. The central angle \( \angle AOB \) is the angle formed by the two radii of the circle that intersect A and B at points O, A, and B.
The measure of a central angle \(^1\) is defined as the angle that it intercepts on the circumference of the circle, and is equal to the ratio of the length of the intercepted arc to the radius of the circle. We can express this relationship mathematically as: $$ \small \text{Measure of central angle } \angle AOB =\frac{ \text{Length of intercepted arc AB}}{ \text{Radius of circle}} $$

We can also express this formula in terms of the central angle's degree measure. Since the circumference of a circle is given by \( 2 \pi r \), where \(r\) is the radius of the circle, and there are 360 degrees in a full circle, we have:
\( \text{Length of intercepted arc AB } = \frac{\theta }{360^\circ} (2 \pi r) \) , here \( \theta \) is the degree measure of the central angle. Substituting this into the formula for the measure of the central angle, we get: $$ \small \text{Measure of central angle} \angle AOB = \frac{\theta }{360^\circ}(2r) = \frac{ \theta }{180^\circ} r $$

This formula is particularly useful when we know the radius of the circle and the degree measure of the central angle, and we want to find the length of the intercepted arc or the measure of the angle that subtends the arc.

The measure of a central angle \(^2\) is equal to the measure of the arc it intercepts. This relationship can be expressed mathematically as: \(\theta = \frac{s}{r} \), where \(\theta \) is the measure of the central angle in radians, \(s\) is the length of the arc intercepted by the angle, and \(r\) is the radius of the circle.
For example, if the radius of a circle is \(r=5\) and an arc of length \(s=3\) intercepts a central angle, the measure of the angle can be found using the formula: \(\theta = \frac{s}{r}=\frac{3}{5} \)
So the measure of the central angle is \( \theta =0.6 \text{radians} \).

Arc of a circle is a portion of the circumference of a circle. It is defined by two endpoints on the circle and is the shortest path between them. The length of an arc can be found using the formula: \(s = r \theta \) , where \(s\) is the length of the arc, \(r\) is the radius of the circle, and \(\theta \) is the measure of the central angle in radians.
For example, if the radius of a circle is \(r=2\) and the central angle intercepts an arc of length \(s=3\), the measure of the angle can be found using the formula: \(\theta = \frac{s}{r}=\frac{3}{2} \)
So the measure of the central angle is \( \theta = 1.5 \) radians, and the length of the arc is: \(s = r \theta = 2 \cdot 1.5 =3 \)
Thus, the arc has a length of 3 units.

Let's consider a circle with center \(O\) and radius \(r\). Suppose that we have two points \(A\) and \(B\) on the circle such that \(A\) and \(B\) are not diametrically opposite points (that is, they do not lie on a line passing through the center of the circle). The arc of the circle that is intercepted by these two points is the portion of the circle's circumference that lies between \(A\) and \(B\), including \(A\) and \(B\) themselves.
The length of an arc of a circle is given by the formula: \( \text{Length of arc } AB = \frac{ \theta }{360^\circ } (2 \pi r) \) , where \( \theta \) is the degree measure of the central angle that subtends the arc AB. This formula follows from the fact that the ratio of the arc length to the circumference of the circle is equal to the ratio of the angle that the arc subtends to the full angle of the circle (which is 360 degrees).
Alternatively, we can rearrange the formula to find the degree measure of the central angle that subtends an arc of length s on a circle with radius \(r\): $$ \text{Degree measure of central angle} = \frac{s}{r} \cdot \frac{180^\circ}{ \pi } $$

in addition to length, arcs of circles can also be measured in terms of their angle measure, which is the degree measure of the central angle that subtends the arc. If we know the radius of the circle and the angle measure of the central angle that subtends an arc, we can find the length of the arc using the formula above.
It is important to note that there are two types of arcs on a circle: minor arcs and major arcs. A minor arc is an arc that measures less than 180 degrees, while a major arc is an arc that measures greater than 180 degrees. A semicircle is a special case of a major arc that measures exactly 180 degrees.

Chord of a circle☰

A chord of a circle is a straight line segment that connects two points on the circumference of the circle. The endpoints of the chord are called the chord's endpoints.
The length of a chord of a circle is given by the formula \(^1\): $$ \text{Length of chord } AB =2r sin(\frac{\theta}{2}) $$ where \(r\) is the radius of the circle, \(AB\) is the length of the chord, and \( \theta \) is the degree measure of the central angle that subtends the chord. This formula can be derived using the Law of Sines, which states that in any triangle \(ABC\), the ratio of the sine of an angle to the length of the opposite side is constant: $$ \frac{sin \angle A}{AB} =\frac{sin \angle B}{BC}=\frac{sin \angle C}{AC} $$ If we let \( \angle A \) be the central angle that subtends chord \(AB\), then \( \angle A \) is also the angle that is opposite side \(AB\) in triangle \(AOB\), where \(O\) is the center of the circle and \(A\) and \(B\) are points on the circumference of the circle. Therefore, we can write: $$ \frac{sin(\frac{\theta }{2})}{r} = \frac{sin(\frac{AB}{2r})}{1} $$ Solving for \(AB\), we get: $$ AB = 2r sin(\frac{\theta }{2} ) $$ Another formula for finding the length of a chord of a circle is by using the perpendicular distance from the center of the circle to the chord. Let the chord be \(AB\) and the center of the circle be \(O\). Let the perpendicular distance from \(O\) to \(AB\) be \(h\), and let the length of the chord be \(AB\). Then the length of the chord is given by: $$ \text{Length of chord }AB=2 \sqrt{r^2-h^2} $$ where \(r\) is the radius of the circle.

This formula allows us to find the length of a chord of a circle if we know the radius of the circle and the perpendicular distance from the center of the circle to the chord. Conversely, if we know the length of a chord and the radius of the circle, we can use this formula to find the perpendicular distance from the center of the circle to the chord: $$ h = \sqrt{r^2 - (\frac{AB}{2})^2 } $$ There are several theorems that are related to chords of circles:

• The perpendicular bisector of a chord passes through the center of the circle. This means that if we draw a line that is perpendicular to the chord and passes through the midpoint of the chord, that line will pass through the center of the circle.
• If two chords of a circle intersect, the product of the segments of one chord is equal to the product of the segments of the other chord. This theorem is known as the intersecting chords theorem. Specifically, if two chords AB and CD intersect at a point E, then: \(AE \cdot EB = CE \cdot ED \)
• If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. This theorem is known as the perpendicular chord bisector theorem. Specifically, if a diameter of a circle is perpendicular to a chord AB, then the diameter bisects AB at its midpoint M, and the arc of the circle intercepted by the chord AB is also bisected by the diameter.
• If two chords of a circle are equal in length, then they are equidistant from the center of the circle. This theorem is known as the chords equidistant from the center theorem. Specifically, if chords AB and CD are equal in length, and O is the center of the circle, then \(OA=OB=OC=OD\)

An angle subtended inside a circle ☰

An angle subtended inside a circle is an angle formed by two intersecting chords, two intersecting secants, or a chord and a tangent, where the vertex of the angle is on the circumference of the circle. The size of the angle is determined by the position of its vertex relative to the center of the circle and the lengths of the chords or secants involved.
The angle subtended by an arc is defined as the angle formed by the two radii that intersect the endpoints of the arc. This angle is also called the central angle, and its measure is equal to the measure of the arc it subtends. That is, if arc \(AB\) has a measure of \(m\) degrees, then the central angle formed by radii \(OA\) and \(OB\) has a measure of \(m\) degrees as well.
Another type of angle subtended inside a circle is an inscribed angle. An inscribed angle is an angle formed by two chords that intersect on the circumference of the circle. The measure of an inscribed angle is half the measure of the arc it subtends. That is, if arc \(AB\) has a measure of \(m\) degrees, then the inscribed angle formed by chords \(AC\) and \(BC\) has a measure of \( \frac{m}{2} \) degrees.
A theorem related to inscribed angles is the inscribed angle theorem, which states that if an angle inside a circle is subtended by a chord, then the angle is half the measure of the arc it subtends. Specifically, if chord \(AB\) subtends arc \(CD\) and angle \(AOC\) is an inscribed angle, then: $$ \angle AOC =\frac{1}{2} \angle ACB = \frac{1}{2} arc CD $$ where \(arc CD\) is the measure of \(arc CD\).

Another theorem related to angles subtended inside a circle is the angle formed by a tangent and a chord theorem. This theorem states that the measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. Specifically, if chord \(AB\) is intersected by tangent line \(PQ\) at point \(P\), and if arc \(ACB\) is the intercepted arc, then: $$ \angle APB =\frac{1}{2} arc ACB $$ where \(arc ACB\) is the measure of \(arc ACB\).

These theorems can be used to solve problems involving angles subtended inside a circle. For example, given the length of a chord and the radius of the circle, we can use the chord length formula and the inscribed angle theorem to find the measure of an inscribed angle or the measure of the intercepted arc. Similarly, given the length of a tangent and the radius of the circle, we can use the Pythagorean theorem and the angle formed by a tangent and a chord theorem to find the length of a chord or the measure of the intercepted arc.

Tangent of a Circle ☰

A tangent of a circle is a straight line that intersects the circle at exactly one point, called the point of tangency. The tangent line is perpendicular to the radius that intersects the point of tangency. Tangent lines play an important role in geometry and have several important properties and theorems associated with them.
One important theorem related to tangents of a circle is the tangent-chord theorem, which states that when a tangent and a chord intersect at a point on the circle, the measure of the angle formed by the tangent and the chord is equal to half the measure of the intercepted arc. Specifically, if the tangent line intersects the chord at point \(P\), and if arc \(ACB\) is the intercepted arc, then: $$ \angle APB = \frac{1}{2} arcACB $$ where \(arcACB\) is the measure of \(arc ACB\).

Another important theorem related to tangents of a circle is the secant-tangent theorem, which states that when a secant and a tangent intersect at a point outside the circle, the product of the lengths of the secant and its external segment is equal to the square of the length of the tangent. Specifically, if secant line AB intersects the tangent line at point \(P\), and if the length of the tangent line from \(P\) to the point of tangency is \(x\), then:
\( PA^2 = PB \cdot PC \) where \(PB\) is the length of the secant line from \(P\) to point \(B\) and \(PC\) is the length of the external segment of the secant.
The length of the tangent from a point outside the circle to the point of tangency can be found using the Pythagorean theorem. Specifically, if the distance from the point to the center of the circle is \(r\), and the distance from the point to the point of tangency is \(x\), then: $$ x^2 = r^2 - d^2 $$ where \(d\) is the distance from the point to the center of the circle.

Tangents also have important applications in calculus, where they are used to define the derivative of a function at a point. The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point. This concept is used in many areas of mathematics and science, including physics, engineering, economics, and more.

Secant of a circle ☰

A secant of a circle is a straight line that intersects the circle at two distinct points. A secant line is different from a tangent line, which intersects the circle at only one point.

One important theorem related to secants of a circle is the intersecting secant theorem, which states that when two secant lines intersect inside a circle, the product of the lengths of the segments of one secant is equal to the product of the lengths of the segments of the other secant. Specifically, if secant lines \(AB\) and \(CD\) intersect inside the circle at point \(P\), and if the lengths of the segments are denoted as follows: \( AP = a \)
\( PB = b \)
\( CP = c \)
\( PD = d \)
then: \( a \cdot b = c \cdot d \)

Another important theorem related to secants of a circle is the secant-tangent theorem, which states that when a secant and a tangent intersect at a point outside the circle, the product of the lengths of the secant and its external segment is equal to the square of the length of the tangent. Specifically, if secant line \(AB\) intersects the tangent line at point \(P\), and if the length of the tangent line from \(P\) to the point of tangency is \(x\), then:
\( PA^2 = PB \cdot PC \) , where \(PB\) is the length of the secant line from \(P\) to point \(B\) and \(PC\) is the length of the external segment of the secant.

The length of a secant line can also be found using the Pythagorean theorem. Specifically, if the distance from the center of the circle to the point of intersection of the secant and the circle is \(r\), and the lengths of the segments of the secant are denoted as follows:
\( AP = a \)
\( PB = b \)
then: \( (a+b)^2 = 4r^2 - (a-b)^2 \)

Angle between the tangents and the secants ☰

Angle between the tangent and the secant:
When a tangent and a secant line intersect outside a circle, the angle between them is equal to half of the difference between the measure of the intercepted arc and 90 degrees. In other words, if a tangent line intersects a circle at point \(A\), and a secant line intersects the circle at points \(B\) and \(C\), with \(B\) outside the circle and \(C\) inside the circle, then the angle between the tangent line and the secant line at point \(A\) is given by: $$ \angle BAC= \frac{1}{2} ( \angle BOC - 90^\circ ) $$ where \( \angle BOC\) is the measure of the intercepted arc.

Angle between two tangents:
When two tangent lines are drawn to a circle from an external point, the angle between the tangent lines is equal to the half of the difference between the measures of the intercepted arcs. Specifically, if two tangent lines are drawn to a circle at points \(A\) and \(B\), and an external point \(P\) is connected to the center of the circle, then the angle between the tangent lines at the external point \(P\) is given by: $$ \angle APB = \frac{1}{2} ( \angle AOB - 90^\circ ) $$ where \( \angle AOB \) is the measure of the intercepted arc.

Angle between two secants:
When two secant lines are drawn from an external point to a circle, the angle between the two secant lines is equal to half of the difference between the measures of the intercepted arcs. Specifically, if two secant lines are drawn from an external point \(P\) to a circle, intersecting the circle at points \(A\),\(B\),\(C\) and \(D\), then the angle between the two secant lines at the external point \(P\) is given by: $$ \angle APB = \frac{1}{2} ( \angle AOC - \angle BOD) $$ where \( \angle AOC \) and \( \angle BOD \) are the measures of the intercepted arcs.

These formulas can be used to calculate the angles between lines intersecting circles in various ways. For example, in geometry problems involving circles, these formulas can be used to find the angle between a tangent and a secant, or between two tangents, or between two secants. In addition, the formulas can be used in calculus to find the slope of tangent lines and the rates of change of curves.