Chords - Higher
The If the angle between two lines is a right angle, the lines are said to be perpendicular. from the centre of a circle to a A straight line joining two points on a circle is a chord. To divide into two equal sections, cut in half. the chord.
Example
A circle has a The distance from the centre of a circle to its circumference. The plural of radius is radii. of 5 cm. The chord EF is 7 cm.
How far is the Midpoint is the middle of a line segment. It divides the line segment in half. of the chord from the centre of the circle?
Add the radii, OE and OF, to make two right-angled triangles.
FM is half of the length of chord EF.
FM = 3.5 cm
Use triangle OFM and Pythagoras's theorem applies to right-angled triangles. The area of the square drawn on the hypotenuse is equal to the sum of the squares drawn on the other two sides. to calculate the length OM.
\(\text{OF}^2 = \text{FM}^2 + \text{OM}^2\)
\(\text{OM}^2 = 5^2 - 3.5^2\)
\(\text{OM}^2 = 12.75\)
OM = 3.6 cm (to 1 decimal place)
Proof
In the diagram below, AB is the chord of a circle with centre O.
OM is the perpendicular from the centre to the chord.
Angles OMA and OMB are both right-angles.
OA is the The longest side of a right-angled triangle, which is opposite the right angle, is called the hypotenuse. of triangle OAM.
OB is the hypotenuse of triangle OBM.
OA = OB as both are radii of the circle.
OM is common to both triangles.
Therefore, triangles OAM and OMB are Congruent means two shapes are exactly the same shape and exactly the same size. (RHS – right-angle, hypotenuse, side).
Therefore, the remaining sides of the triangles are equal, AM = MB.
So, M must be the mid-point of AB, and the chord has been bisected.