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?
The two radii OE and OF are the The longest side of a right-angled triangle, which is opposite the right angle, is called the hypotenuse. of 2 right-angled triangles.
The distance FM is half of the length of the chord. (The perpendicular from the centre of a circle to a chord bisects the chord.)
FM = 3.5 cm
Use 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.
OF2 = FM2 + OM2
52 = \(3.5^2 + OM^2\)
OM2 = \(5^2 - 3.5^2\)
OM2 = 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.
Look at triangles OAM and OBM.
The hypotenuses (OA and OB) are the same, as they are both the radius of the circle.
OM is common to both triangles.
OMA and OMB are both right angles.
Triangles OAM and OBM are congruent (RHS), so it follows that AM = MB.
Therefore, M is the midpoint of AB, and the chord has been bisected.
More guides on this topic
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- Angles, lines and polygons - AQA
- Loci and constructions - AQA
- 2-dimensional shapes - AQA
- 3-dimensional shapes - AQA
- Circles, sectors and arcs - AQA
- Transformations - AQA
- Pythagoras' theorem - AQA
- Units of measure - AQA
- Trigonometry - AQA
- Vectors - AQA
