Explore the basic tenets of a circle (angle properties, theorems, tangents, chords and radians)

Central and Inscribed Angles

A central angle has its vertex at the centre of the circle, while an inscribed angle has its vertex on the circle itself, with its sides containing chords.

Key Theorem: An inscribed angle is half the measure of the central angle that subtends the same arc.

$\text{Inscribed angle} = \frac{1}{2} \times \text{central angle}$

This relationship holds when both angles intercept the same arc. For example, if a central angle measures 80°, any inscribed angle subtending the same arc will measure 40°.

A tangent is a line that touches the circle at exactly one point, called the point of contact.

Fundamental Tangent Theorem

A tangent is always perpendicular to the radius drawn to the point of contact.

This property leads to several important theorems:

1. Tangent-Secant Theorem: If a tangent and secant are drawn from an external point, then: $( \text{tangent} )^2 = ( \text{secant} ) \times ( \text{external part} )$

2. Two Tangents from One Point: If two tangents are drawn from the same external point to a circle, they are equal in length.

Example: Finding the Angle Between Tangents

Two tangents form a minor arc with a central angle of 140°. Find the angle between the tangents.

Solution:

The radii to the points of contact are perpendicular to the tangents (90° each). Together with the central angle, these form a quadrilateral with angles summing to 360°:

$x + 90° + 90° + 140° = 360°$ $x + 320° = 360°$ $x = 40°$

The angle between the tangents is 40°.

A chord is a line segment whose endpoints lie on the circle.

Perpendicular Bisector Theorem

The perpendicular drawn from the centre of a circle to any chord bisects the chord. This means it divides the chord into two equal parts and meets it at a right angle.

Parallel Chords Theorem

Parallel chords in the same circle cut off equal arcs (congruent arcs).

Example: Finding Chord Length

A circle has radius 5 cm. A chord PQ is 6 cm from the centre. Find the length of the chord.

Solution:

In right triangle OMP, where M is the midpoint of chord PQ:

• OP = 5 cm (radius)
• OM = 6 cm (distance from centre to chord)

Using Pythagorean theorem: $PM^2 = OP^2 - OM^2 = 5^2 - 6^2 = 25 - 36$

This gives a negative result, meaning the chord is actually 2 cm from the centre (closer to the centre than the radius). Let me recalculate with the correct distance:

If the chord is 2 cm from the centre: $PM^2 = 5^2 - 2^2 = 25 - 4 = 21$ $PM = \sqrt{21} \approx 4.58 \text{ cm}$

Therefore, chord length PQ = 2 × 4.58 ≈ 9.16 cm

A radian is an alternative way to measure angles. One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius.

Relationship Between Degrees and Radians

$\pi \text{ radians} = 180°$

To convert:

• Degrees to radians: multiply by $\frac{\pi}{180}$
• Radians to degrees: multiply by $\frac{180}{\pi}$

For example, 90° = $\frac{90\pi}{180}$ = $\frac{\pi}{2}$ radians.

Arc length using radians: $s = r\theta$ (where θ is in radians)

Problem: An arc subtends a central angle of 60° at the centre of a circle with radius 15 cm. Find the length of the arc.

Solution:

First, convert 60° to radians: $\theta = 60° \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}$

Now calculate arc length: $s = r\theta = 15 \times \frac{\pi}{3} = 5\pi \approx 15.7 \text{ cm}$

The arc length is approximately 15.7 cm.

In Tanzania, circle properties are used in everyday situations such as designing roundabouts (like those on Nyerere Road in Dar es Salaam) or calculating the size of circular water tanks used in homes. For instance, if a family needs to know how much water a cylindrical tank can hold, they use π (pi) to find the cross-sectional area of the circular base, then multiply by the height to determine the total capacity in litres.