Theorems with Chords & Tangents (Edexcel IGCSE Maths A (Modular))
Revision Note
Written by: Amber
Reviewed by: Dan Finlay
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Circles & Chords
What is a chord?
A chord is any straight line is a circle that joins any two points on the circumference
Chords of equal length are equidistant (the same distance) from the centre
Circle Theorem: The perpendicular bisector of a chord passes through the centre
If a line through the centre (such as a radius or diameter) goes through the midpoint of chord
it will bisect (cut in half) that chord at right angles to it
To spot this circle theorem on a diagram
look for a radius and see if it intersects any chords
or look to see if you could draw a radius that bisects a chord
When explaining this theorem in an exam you can use either phrase below:
A radius bisects a chord at right angles
The perpendicular bisector of a chord passes through the centre
Examiner Tips and Tricks
Look out for isosceles triangles formed by a chord and two radii
Two angles in the triangle will be equal and there will be at least one line of symmetry
Worked Example
The diagram below shows a circle with centre, O.
Two points, P and Q, lie on its circumference.
The radius of the circle is 6 cm.
Angle OPQ = 40º.
Find the length PQ.
Label the radius on the diagram 6 cm
Draw a line from O to the midpoint, M, of the line PQ
The angle formed between the OM and PQ will be a right angle
Use SOHCAHTOA on triangle OMQ to find the length MQ
Double MQ to find the length PQ
Round to 3 significant figures
PQ = 9.19 cm (3 s.f.)
Circles & Tangents
What is a tangent?
A tangent to a circle is a straight line outside of the circle that touches its circumference at exactly one point
Circle Theorem: A radius and a tangent meet at right angles
If a radius and a tangent meet at a point on the circumference of a circle, the angle formed between them will be 90°
They are perpendicular to each other
When explaining this theorem in an exam you must use the keywords:
A radius and a tangent meet at right angles
Circle Theorem: Tangents from an external point are equal in length
Two tangents from the same external point are equal in length
This means that a kite can be formed by two tangents meeting a circle
The kite below has a vertical line of symmetry
It is formed from two congruent triangles back-to-back
The kite will have two right angles where the tangents meet the radii
You can use Pythagoras and SOHCAHTOA on each of these triangles
Examiner Tips and Tricks
Look for tangents in the exam and draw on the radius at right angles to see if it helps
Worked Example
Find the value of θ in the diagram below.
The lines ST and RT are both tangents to the circle
They meet the two radii on the circumference at the points S and T
Angle TSO = angle TRO = 90°
A radius and a tangent meet at right angles
Use vertically opposite angles to find the value of angle RTS
Angle RTS = 25°
Vertically opposite angles
Mark these angles clearly on the diagram
Angles in a quadrilateral add up to 360°
Use this to form an equation for θ
Angles in a quadrilateral sum to 360º
Simplify
Solve
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