Circles 2

Chord

When any two points A & B on the circumference are joined by a straight line, AB is called a Chord of the circle. The minimum length of a chord is zero. The maximum length of a chord is the Diameter of the circle.

The line joining the mid-point of any chord to the centre of the circle will be perpendicular to the chord.

The center of any circle can be located if any 2 chords of the circle are known. We just have to draw the perpendicular bisectors of both the chords. The point at which they intersect is the centre of the circle.

Extension to Pythagoras Theorem

Greeks understood the Pythagoras theorem in terms of areas - "the area of the square on the hypotenuse will be equal to the sum of the areas of the squares on the other 2 sides"

Area of a square is the square of the side. Similarly, the area of a circle is.

Hence, we can reframe the theorem in terms of circles - the area of the circle/ semicircle on the hypotenuse will be equal to the sum of the areas of the circles/ semicircles on the other 2 sides.

This also gives the idea of drawing 2 circles whose areas will total to the area of a given circle!

Imagine that we draw a right angle triangle and construct 3 circles using the sides as diameters. The area of the circle on the hypotenuse would be equal to the areas of the circles on the other two sides!

Cyclic Quadrilaterals

When we plot 2 points on the circumference, it divides the circle into 2 segments. We saw that the angles in both these segments are supplementary or total to 180.

This result can be generalized as a property of any quadrilateral whose vertices fall on the circumference of circle. We know that the sum of all angles of any quadrilateral is 2 straight angles (360). If two of its opposite angles total to 180, then obviously the sum of the other 2 opposite angles should also be 180!

Such quadrilaterals are called cyclic quadrilaterals. They have the property that their opposite angles are supplementary.

Hexagon in a Circle

One of the easiest constructions was found to be the drawing of a Hexagon inscribed in a circle. The hexagon itself is formed of 6 congruent equilateral triangles. Make the 2 legs of a compass equal to that of the radius. Starting from any point on the circumference, successively mark off points on the circumference using the compass. The sixth point will coincide with the first point.

The six points on the circumference, would form the vertices of a regular hexagon. The hexagon can be divided into 6 equilateral triangles. The perimeter of this hexagon, which is inside the circle is 6r. Hence we can see that the value of π should be more than 3!

Tangent to the Circle

Another idea which emerged was the drawing of a line which is a tangent to the circle. It was also found that at the point of contact a tangent and the radius passing through the point are at right angles.

Periodic Table of Shapes

Mathematicians believe that mathematical shapes can be broken down into more basic components. Over the past decade, scientists from the Imperial College of London have been working to build a periodic table of shapes. Now, with the help of AI and machine learning, they’re making great strides towards achieving this goal.