Edexcel Mathematics (4XMAF) -Unit 2 - 4.6 Circle Properties- Study Notes- New Syllabus
Edexcel Mathematics (4XMAF) -Unit 2 – 4.6 Circle Properties- Study Notes- New syllabus
Edexcel Mathematics (4XMAF) -Unit 2 – 4.6 Circle Properties- Study Notes -Edexcel iGCSE Mathematics – per latest Syllabus.
Key Concepts:
A recognise the terms centre, radius, chord, diameter, circumference, tangent, arc, sector and segment of a circle
B understand chord and tangent properties of circles:
Two tangents from a point to a circle are equal in length
A tangent is perpendicular to the radius at the point of contact
The line from the centre perpendicular to a chord bisects the chord (and converse)
Edexcel iGCSE Mathematics -Concise Summary Notes- All Topics
Parts of a Circle
A circle is a set of points that are all the same distance from a fixed point.
Centre
The fixed point in the middle of the circle.
Radius
A line from the centre to the circle.
Diameter
A straight line passing through the centre joining two points on the circle.
\( \text{Diameter} = 2 \times \text{Radius} \)
Chord
A straight line joining any two points on the circle.
Circumference
The distance around the circle (the perimeter of the circle).
Tangent
A straight line that touches the circle at exactly one point.
Arc
A part of the circumference of a circle.
Sector
A region bounded by two radii and an arc.
Segment
A region bounded by a chord and an arc.
Example 1:
A circle has radius 6 cm. Find the diameter.
▶️ Answer/Explanation
\( 2 \times 6 = 12 \text{ cm} \)
Conclusion: 12 cm.
Example 2:
Name the line touching the circle at one point only.
▶️ Answer/Explanation
Tangent.
Conclusion: Tangent.
Example 3:
Which part of a circle is bounded by two radii and an arc?
▶️ Answer/Explanation
Sector.
Conclusion: Sector.
Chord and Tangent Properties of Circles
1. Tangent is Perpendicular to the Radius
A tangent touches the circle at exactly one point.
At the point of contact:
The radius is perpendicular to the tangent.
Angle = \( 90^\circ \)
2. Two Tangents from the Same External Point are Equal
If two tangents are drawn from a point outside the circle to the circle, the lengths of the tangents are equal.
\( PA = PB \)
3. Perpendicular from Centre to a Chord Bisects the Chord
If a line from the centre of the circle meets a chord at \( 90^\circ \), it cuts the chord into two equal parts.
\( AX = XB \)
Converse
If a line from the centre bisects a chord, then the line is perpendicular to the chord.
Example 1:
A radius meets a tangent at the point of contact. Find the angle between them.
▶️ Answer/Explanation
Tangent ⟂ radius.
\( 90^\circ \)
Conclusion: \( 90^\circ \).
Example 2:
Two tangents from a point to a circle measure 7 cm each. Find the length of the second tangent if one is 7 cm.
▶️ Answer/Explanation
Tangents from the same point are equal.
Conclusion: 7 cm.
Example 3:
A perpendicular from the centre meets a chord of length 10 cm. Find the length of each half.
▶️ Answer/Explanation
The perpendicular bisects the chord.
\( 10 ÷ 2 = 5 \text{ cm} \)
Conclusion: Each half is 5 cm.