3.4.1 Angles at center & Semicircles
A circle is a two-dimensional geometric shape that consists of all the points in a plane that are a fixed distance (called the radius) from a given point (called the center).
The main parts of a circle are:
Center: The point that is equidistant from all points on the circle.
Radius: The distance from the center of the circle to any point on the circle.
All radii of a circle are equal.
Diameter: The distance across the circle passing through the center.
The diameter is equal to two times the radius.
Chord: A line segment that connects two points on the circle.
A diameter is the longest chord in a circle.
Tangent: A line that touches the circle at only one point, called the point of tangency.
Secant: A line that intersects the circle at two distinct points.
Arc: A portion of the circumference of the circle.
It can be measured in degrees or radians.
Sector: A region of the circle that is bounded by two radii and an arc.
The measure of a sector is the central angle that subtends it, measured in degrees or radians.
Segment: A region of the circle that is bounded by a chord and an arc.
Understanding the parts of a circle is important in geometry and other mathematical disciplines, as well as in engineering, physics, and other fields that involve circular objects and shapes.
Angle at center:
An angle at the center of a circle is an angle whose vertex is located at the center of the circle and whose sides intersect the circumference of the circle.
The measure of an angle at the center of a circle is equal to twice the measure of the corresponding angle at the circumference that intersects the same arc. In other words, if two angles are formed by two intersecting chords or tangents of a circle at the same endpoint of an arc, the angle at the center is twice the angle at the circumference.
The “triangle parts” can also overlap in this theorem.
This property is important in solving problems involving arcs, chords, tangents, and angles in circles.
It is commonly used in calculations involving the length of arcs, the area of sectors, and the angles formed by intersecting chords or tangents.
Angle in a semicircle:
In a semicircle, an angle formed by any two points on the circumference and the center of the circle will always measure \( 90 \) degrees (or \( \frac{π}{2} \) radians).
This can be proved using the fact that a line drawn from the center of a circle to a point on its circumference is always perpendicular to the tangent of the circle at that point.
Since a semicircle is just half of a circle, the angle formed by any two points on the circumference of a semicircle and the center of the circle will be half of the angle formed by the same two points and the center of the corresponding full circle, which is always \( 180 \) degrees (or \( π \) radians).
Therefore, the angle in a semicircle will always measure \( 90 \) degrees (or \( \frac{π}{2} \) radians).
Chord:
A chord is a straight line that connects two points on the circumference of a circle. The two points are called the endpoints of the chord.
A chord is different from a diameter, which is also a line that connects two points on the circumference of a circle but passes through the center of the circle.
A chord may or may not pass through the center of the circle.
The length of a chord can be calculated using the distance formula or by applying the Pythagorean theorem.
If the length of the chord is known, the distance between the chord and the center of the circle can be calculated using the formula for the distance from a point to a line.
Chords have several important properties that are used in geometry and trigonometry, including the chord-chord product theorem, which relates the lengths of two chords that intersect inside a circle, and the inscribed angle theorem, which relates the angle subtended by a chord and an arc of a circle.
Perpendicular bisector of a chord:
The perpendicular bisector of a chord is a line that intersects the chord at a right angle and also passes through the center of the circle that contains the chord.
The perpendicular bisector divides the chord into two equal parts and is equidistant from the endpoints of the chord.
The perpendicular bisector of a chord can be used to find the center of a circle that contains the chord.
To do this, you can draw the perpendicular bisector of the chord and locate the midpoint of the bisector.
This midpoint will be the center of the circle.
Conversely, if you know the center of a circle and the endpoints of a chord, you can construct the perpendicular bisector of the chord by drawing a line that passes through the center of the circle and the midpoint of the chord.
The perpendicular bisector of a chord has several important applications in geometry, such as finding the radius of a circle, constructing regular polygons inscribed in a circle, and determining the position of a point relative to a circle.
It is a fundamental concept in the study of circles and their properties.
Tangent:
A tangent is a straight line that touches a curve at a single point, without crossing through the curve at that point. The point where the tangent touches the curve is called the point of tangency.
Tangents are commonly used in the study of circles.
If a line is tangent to a circle, it intersects the circle at only one point and is perpendicular to the radius that intersects that point. This means that the tangent line is at a right angle to the radius at the point of tangency.
The slope of a tangent line can be used to calculate the derivative of a function in calculus.
The derivative represents the rate of change of the function at a particular point, and is defined as the limit of the slope of a secant line that passes through two points on the curve as the points get closer and closer together.
Tangents have many practical applications in fields such as engineering, physics, and computer graphics.
For example, the tangent to a curve can be used to calculate the velocity of a moving object at a particular point in time, or to determine the path of a light ray through a curved surface.
Tangent is perpendicular to radius:
A key property of tangents and radii is that the tangent line to a circle at a given point is always
perpendicular to the radius that connects that point to the center of the circle. This means that if you draw a line from the center of the circle to the point of tangency, it will be perpendicular to the tangent line at that point.
The relationship between the radius and tangent of a circle has many practical applications in fields such as trigonometry, geometry, and physics.
For example, in optics, the tangent of the angle between a ray of light and the normal to a curved surface is used to calculate the angle of refraction, which determines the direction in which the light will bend as it passes through the surface.
Cyclic quadrilateral:
A cyclic quadrilateral is a quadrilateral (a four-sided polygon) whose vertices all lie on a single circle. In other words, if you draw the circle that passes through all four vertices of the quadrilateral, the quadrilateral is said to be cyclic.
One of the interesting properties of a cyclic quadrilateral is that the opposite angles are supplementary, meaning that the sum of two opposite angles is equal to \( 180 \) degrees.
This is known as the “opposite angles” or “interior angles” theorem for cyclic quadrilaterals.
Another property of cyclic quadrilaterals is that the measures of the two pairs of opposite angles add up to \( 360 \) degrees.
This means that if you know the measures of three of the angles of a cyclic quadrilateral, you can find the measure of the fourth angle using this property.
Cyclic quadrilaterals have many applications in geometry and trigonometry, and they are often used in problem-solving and proofs.
For example, the cyclic quadrilateral property can be used to solve problems involving inscribed angles, tangents, and secants in circles.
The theorem only applies to cyclic quadrilaterals.
Do not be fooled by other quadrilaterals in a circle.
The diagram below depicts a common scenario that is NOT a cyclic quadrilateral.
Angle at circumference subtended by the same arc:
When two or more angles are formed by connecting two points on the circumference of a circle with a third point inside the circle, and they all subtend the same arc, then they are equal in measure. This is known as the angle at circumference subtended by the same arc theorem, or simply the inscribed angle theorem.
To state it more formally, the inscribed angle theorem says that:
In a circle, if two chords (lines that connect two points on the circumference) intersect at a point within the circle, then the angles between the chords at the point of intersection are equal if and only if the chords subtend equal arcs on the circumference.
This theorem is often used to solve problems involving circles, such as finding the measure of an angle or the length of a chord.
Alternate segment theorem:
The alternate segment theorem is a theorem in geometry that relates the angles formed by a chord and a tangent line to a circle.
The theorem states that:
The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the opposite segment.
In other words, if a tangent line is drawn to a circle from a point on the circumference, and a chord is drawn from that same point to another point on the circle, then the angle between the tangent and the chord (at the point of contact) is equal to the angle formed by the chord in the opposite segment of the circle.
This theorem is often used to solve problems involving tangents and chords in circles, such as finding the measure of an angle or the length of a chord.
Test yourself
Question 1:
In the diagram , \( A, B, C \) and \( D \) lie on a circle, center \( O \).
\( EA \) is a tangent to the circle at \( A \).
Angle \( EAB=61° \) and angle \( BAC=55° \).
Find angle \( BAO \).
[1]
Find angle \( AOC \).
[1]
Find angle \( ABC \).
[1]
Question 2:
The points \( A, B, C, D \) and \( E \) lie on the circle.
\( PAQ \) is a tangent to the circle at \( A \) and \( EC=EB \).
Angle \( ECB=80° \) and angle \( ABE=40° \).
Find the values of \( v, w, x, y \) and \( z \).
[5]
Question 3:
In the diagram, \( K, L \) and \( M \) lie on the circle, center \( O \).
Angle \( KML=2x° \) and reflex angle \( KOL=11x° \).
Find the value of \( x \).
[3]
Question 4:
\( A, B, C \) and \( D \) are points on the circle, center \( O \).
Angle \( ABD=21° \) and \( CD=12 \)cm.
Calculate the area of the circle.
[5]
Question 5:
\( A, B, C \) and \( D \) are points on the circumference of a circle, center \( O \).
\( FDE \) is a tangent to the circle.
Show that \( y-x=90 \)
You must give a reason for each stage of your working.
[2]
Dylan was asked to give some possible values for \( x \) and \( y \).
He said ” \( y \) could be \( 200 \) and \( x \) could be \( 110 \), because \( 200-110=90 \)”
Is Dylan correct \( ? \)
You must give a reason for your answer.
[1]