Circle Theorems: Simplified for students
Circle Theorems in Mathematics can be so brain-wrecking and disturbing sometimes. It’s even more annoying with the angles and different triangles that pop up.
But the truth is complexity is the enemy of understanding. When circle theorems are stripped of their ambiguity, it seem a lot more naked and understandable.
That’s what I’m here to help with today. Simplifying Circle Theorems to show that they are nothing more than…Lines.
Am I telling you that after all your years of difficulty with circle theorems, they are nothing more than just lines? Yes.
As you start drawing lines in the circle and jointing one point to another, patterns begin to form, the picture gets clearer, you see the angles, the triangles, segments and so much more. It all begins with…a line.
The most important feature in circle theorems is the radius (sing.), radii (plu.). It is from the radius we can go on to deduce various circle theorems there is to know. These lines have specific names to them, so you should know some of these circle theorems terms before we begin to unravel the theorems themselves.
Some Most Used Line-Terms In Circle Theorems
These terms will pop up here and there during my explanation. I can’t have you switching back and forth in your browser to understand what I mean.
1. Radius
The radius is a straight line that starts from the centre of a circle to any point on the circumference of the circle. The circumference is the body of the circle, stay with me.
The formula for calculating the radius of a circle is Diameter divided by 2(D/2). The length of the radius is half the diameter of the circle.
That is:
Radius = Diameter * 1/2
Radius = Diameter/2(D/2).
The radius is an essential measurement in circle theorems and plays an important role in various formulas.
2. Diameter
Unlike the radius, the diameter doesn’t start at the centre of the circle but passes through the centre. A diameter is a line that starts from any point on the circumference to another point on the circumference.
Note: A Diameter is not a chord. It must pass through the centre of the circle. The diameter is the longest distance across the circle and is twice the length of the radius.
A mathematics guru like yourself should be able to express the formula for that statement.
Diameter = 2*r
Therefore Diameter = 2r
3. Chord
A chord is a line that cuts a circle into two unequal halves. It does not pass through the middle of the circle. The chord touches both ends of a circumference but would not pass through the middle resulting in the unequal halves.
4. Tangent
A tangent is a straight line that touches a circle at exactly one point. This point is called the point of tangency. Unlike a chord, which intersects the circle at two points, a tangent only makes contact at a single point and does not cross into the circle's interior.
The tangent line is perpendicular to the radius of the circle at the point of tangency.
5. Segment
A segment is a region bounded by a chord and the arc that the chord forms. You can either have a minor segment or a major segment.
A minor segment is the shorter arc formed by a chord while the major segment… well, you know what it is.
These are some of the most used line terms in Circle theorems. When you see them pop up, you know what they mean already or you could scroll back. Speaking of Circle Theorems, one thing you must pay attention to compulsorily are isosceles triangles. They pop up more than you’d believe when we start analyzing.
An isosceles triangle has two equal sides, they are gotten in circles by drawing two radii like this...
Alright, let’s talk about the circle theorems we have.
Circle Theorems
These are the theorems we will be proving today.
1. Angle at the centre is twice the angle at the circumference
Is the angle at the centre of a circle twice the angle of the circumference? Well, it’s proving time.
To do this. draw in your circle two radii that meet at a point on the circumference. You should have something like this.
Now, Draw lines to connect A to D and B to C. Once you draw these lines, you will have four triangles.
ACD
ABC
ABD
DCB
Triangles (ACD, ABC and ABD)are all isosceles triangles because they have two equal sides.
In triangle ABC, the angle at A is 180 - 2oS (180o is the total number of angles in a triangle). The largest triangle DCB has;
(u+s) + (u+t) + (t+s) = 180o (Total number of angles in triangle DCB
(open the bracket)
u+u+t+t+s+s = 180o
2u +2t+2s = 180o
2u+2t = 180-2s
That proves the theorem (Angle at the centre is twice the angle at the circumference).
2. The angle in a semicircle is a right angle
This show an application of the semicircle theorem, where the central angle equals 180°. To show this without relying on the theorem, we can proceed as follows:
Draw a line from point (A) to point (C), creating a radius of the circle.
This forms three triangles: (ABC), (ACD), and the larger triangle (BCD). Both (ABC) and (ACD) are isosceles triangles.
First, observe that (a + b = 180o) since they form a straight line.
For (ABC):
[ b + 2s = 180o ] (since the sum of angles in a triangle is (180o)....1
For (ACD)
[ a + 2t = 180o ] (since the sum of angles in a triangle is (180o)..... 2
Adding equations (1) and (2), we get:
[ b + 2s + a + 2t = 360o ]
Since we know (a + b = 180o), we can substitute:
[ 2s + 2t + 180o = 360o ]
[ 2s + 2t = 180o ]
[ s + t = 90o ]
This confirms that the angle in the semicircle is indeed a right angle.
3. Angles in the same segment are equal
The way we demonstrate this is by applying the first theorem. According to this theorem, the angle at (C) (on the circumference) is half the angle at (A) (the center of the circle).
Similarly, the angle at (D) (also on the circumference) is half the angle at (A). Also, the angle at (C) is equal to the angle at (D). This relationship helps to confirm that the angle subtended by the diameter of a semicircle is a right angle.
4. Opposite angles in a cyclic quadrilateral sum to 180°
A cyclic quadrilateral is a quadrilateral where all the vertices are on the circumference of a circle.
Angle B + angle D = angle C + angle E = 180°
Draw radius from the centre to each corner of the quadrilateral. This gives us four isosceles triangles: ABC, ACD, ADE and ABE.
We know that the sum of the interior angles of a quadrilateral is 360°.
Given this, we have:
Angle B Angle C Angle D Angle E
(x+u) + (u+v) + (v+w) + (x+w) = 360°
2u + 2v + 2w + 2x = 360°
u + v + w + x = 180°
Thus,
Angle B + Angle D
(u + x) + (v + w) = 180°
and
Angle C + Angle E
(u + v) + (x + w) = 180°
5. The angle between the chord and the tangent is equal to the angle in the alternate segment
Draw radii from the centre of the circle to each of the points B, C and D.
This forms three isosceles triangles: ABC, ABD, and ACD.
We want to demonstrate that ( a = u + v ).
In the larger triangle ∆BCD, we have:
(u + w) + (v + w) + (u + v) = 180o(sum of angles in a triangle)
Simplifying this, we get:
2u + 2v + 2w = 180o
u + v + w = 90o
Additionally, we know:
a = 90o(tangent and radius meet at 90°)
Substituting this into (1) gives us:
u + v = 90° - w
u + v = a.
That brings us to the end of this, Ladies and Gentlemen. I hope these have been simplified enough for you. Circle theorems hide behind a cluster of lines but when you can see them distinctly like I do, you are good to go!
Conclusion
That's all I've got for you today. Circle Theorems employs geometry in it. These theorems can make you question your math skills. That's because, on the surface, it seems much. But when you peel those layers off,
They are just lines!
