Tangents to a Circle

A tangent to a circle is perpendicular to the radius at the point of tangency. A tangent intersects a circle at one point. Point A is the point of tangency to circle N.

As the tangent AP is perpendicular to radius AO, angle PAO is 90 degrees. Therefore any other tangent to the circle N is 90 degrees to the raidius at any points.

Example

In the figure on the right, P is a point outside the circle, with centre O, PA and PB are two tangents drawn from P to touch the circle at A and B respectively. We can find that
i) AP = BP
ii) ÐAPO = ÐBPO
iii) ÐAOP = ÐBOP

ÐOAP = ÐOBP = 90° (tan ⊥ rad.)
△AOP and △BOP are congruent (RHS Property)
AP = BP
ÐAPO = ÐBPO and ÐAOP = ÐBOP

We can conclude that:

a) tangents drawn to a circle from an external point are equal

b) the tangents subtend equal angles at the centre

c) the line joining the external point to the centre of the circle bisects the angle between the tangents.

In real life application, proffessional billard players use this theory in some of their skills. They use Tangents of the cue ball to find out the direction of where the ball will move.

Done by Ng Ding Yuan

The Perpendicular Bisector of a Chord

A chord is a line segment that passes through a circle joining any 2 points that touches the circumference of a circle. When a chord is drawn through the centre of the circle, it divides the circle into two identical parts called the semi-circle.

When there is a perpendicular bisector CD of the line segment AB, it is a line segment perpendicular to AB that passes through the midpoint M.

The perpendicular bisector of a line segment can be constructed by drawing a line segment perpendicular which passes through the centre, perpendicular to an existing line segment. The green line segment SR is the existing line segment while the green line AO is the perpendicular bisector of the line segment SR. Point O is the centre of the circle and angle SAO is a right-angle. Thus, line AO is perpendicular to line SR, making it a perpendicular bisector.

One of the various applications using Perpendicular bisector of a chord in real-life is to locate the centre of a circular structure or object. For example, the singapore flyer. Have you wondered how did the architects find the exact centre of such a huge circular stucture? Now you know that they could have used the Perpendicular bisectors of the Singapore Flyer's cords to find the flyer's axis. The multiple perpendicular bisectors

crisscrossing one another will show the centre of the flyer where the perpendicular bisectors meet.

Done by Benedict Tan

Concept of Circumference of a Circle

As you know, a circle is a shape with all points the same distance from the center. When the distance around a circle is measured and divided by the distance across the same circle through the center, you will always come close to the value aproximately 3.14159265358979323846... The unit used to represent this value is π. Even up till now, mathematicians are calculating the exact number of π as the value goes on forever.

The distance around a circle is called the circumference of the circle, while the distance across a circle is called the diameter. To find the circumference of a circle, you have to take the diameter and multiply it by the value of π which is commonly simplified to 3.142. This is possible as for any circle, if you divide the circumference by the diameter, you can get the value of 3.142 rounded to 3 decimal places. Thus the formula of finding the circumference of a circle is d x π = c

The radius of a circle is the distance for the center of the circle to any point on the circumference. Therefore, two radi (plural for radius) of a circle placed end to end of each other will form the diameter of a circle. Thus producing the formula,
d = 2r , where d is the diameter and r is the radius.

A real-life application using the circumference of a circle is the measuring of the circumference of your dinning plate. To find the circumference of circular objects in real-life like your dinning plate, you can use a measuring instrument to find the diameter of the plate and apply the d x π = c formula. For example if your plate is 9 inches in diameter, taking 9 x π , you can find out that your plate is actually around 28.3 inches in diameter.

Done by Danny Ng