11.2 Circle Theorems: Chords, Tangents, Arcs & Segments

Circle Theorems: Chords, Tangents, Arcs & Segments

Six of the ten topics NTS names in the Geometry block are circle topics: Chords of a circle, Tangent to a circle, Chords and Arcs, Angles in a segment of a circle, Practical geometry circles, and the circle constructions inside Practical Geometry. That weighting is unusual for an aptitude test and reflects the Pakistani secondary geometry syllabus. Circle items are almost always solved by naming the right theorem, not by heavy computation.

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Vocabulary First

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Chord Theorems

1. The perpendicular from the centre to a chord bisects the chord — and conversely, the line from the centre to a chord's midpoint is perpendicular to it.
2. Equal chords are equidistant from the centre, and chords equidistant from the centre are equal.
3. The longer the chord, the nearer it lies to the centre. The diameter, at distance zero, is the longest.

Theorem 1 is the workhorse: it creates a right triangle whose legs are the perpendicular distance $d$ and half the chord, with the radius as hypotenuse.

$r^{2} = d^{2} + \left(\frac{\text{chord}}{2}\right)^{2}$

Worked example. A chord of length 24 cm lies in a circle of radius 13 cm. How far is it from the centre?

$13^{2} = d^{2} + 12^{2} \;\Rightarrow\; 169 = d^{2} + 144 \;\Rightarrow\; d = 5\text{ cm}$

Using the full chord length of 24 instead of the half-chord is the designed error and yields no real answer at all.

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Angles in Arcs and Segments

Worked example. An arc subtends 130° at the centre. What angle does it subtend at a point on the major arc?

$\text{Angle at circumference} = \frac{130°}{2} = 65°$

Worked example. In a cyclic quadrilateral $PQRS$, $\angle P = 78°$. Then $\angle R = 180° - 78° = 102°$. Note that this says nothing about $\angle Q$ and $\angle S$ individually — only that they too sum to 180°.

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Tangent Theorems

1. A tangent is perpendicular to the radius at the point of contact. This creates a right triangle and is the entry point to almost every tangent problem.
2. Two tangents from an external point are equal in length, and the line from that point to the centre bisects the angle between them.
3. Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

Worked example. From a point 17 cm from the centre of a circle of radius 8 cm, a tangent is drawn. Find its length.

Radius, tangent and the line to the centre form a right triangle with the right angle at the point of contact:

$\text{tangent}^{2} = 17^{2} - 8^{2} = 289 - 64 = 225 \;\Rightarrow\; \text{tangent} = 15\text{ cm}$

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Power of a Point

Worked example. Two chords cut each other inside a circle. One is divided into 6 cm and 8 cm, the other into 4 cm and $x$ cm.

$6 \times 8 = 4 \times x \;\Rightarrow\; x = 12\text{ cm}$

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Arc Length and Sector Area

For a central angle $\theta$ in degrees and radius $r$:

$\text{Arc length} = \frac{\theta}{360}\times 2\pi r \qquad\qquad \text{Sector area} = \frac{\theta}{360}\times \pi r^{2}$

$\text{Segment area} = \text{sector area} - \text{area of the triangle formed by the two radii and the chord}$

Worked example. A sector of a circle of radius 14 cm has a central angle of 90°. Its arc length is $\frac{1}{4}\times 2\pi(14) = 7\pi \approx 22$ cm, and its area is $\frac{1}{4}\pi(196) = 49\pi \approx 154$ cm².

Method that solves most circle items: draw the radius to every labelled point. Radii are equal, so every radius you draw creates an isosceles triangle with two known base angles — and the missing angle usually falls out immediately.