Arcs and chords of a circle

What you need: the minor arc equals the angle at the centre; the major arc is $360^\circ$ minus the minor arc.

Steps:

1. Read off the minor arc, then subtract it from $360^\circ$.

What you need: $AB=2\sqrt{R^2-OH^2}$.

Steps:

1. Find half the chord, $HA$, then double it.

What you need: the foot of the perpendicular, $H$, is the midpoint of $AB$.

Steps:

1. Find $HA=12$ cm.
2. Use Pythagoras' theorem in the triangle $OHA$.

What you need: a triangle with three equal sides is equilateral.

Steps:

1. Show that the triangle $OAB$ is equilateral and deduce $\angle AOB$.

What you need: the minute hand turns $6^\circ$ per minute; the hour hand turns $30^\circ$ per hour.

Steps:

1. Multiply the number of minutes by $6^\circ$.
2. The hour hand turns for $\dfrac{45}{60}$ of an hour.

What you need: the altitude $OH$ of the isosceles triangle $OAB$ is also an angle bisector.

Steps:

1. Deduce that $\angle AOH=60^\circ$.
2. Find $HA=OA\cdot\sin60^\circ$, then double it.

What you need: in an isosceles triangle, the median from the apex is also an altitude and an angle bisector.

Steps:

1. Use the isosceles triangle $OAB$, then Pythagoras' theorem.
2. Find $\tan\angle AOH$, then $\angle AOB=2\angle AOH$.
3. Subtract the minor arc from $360^\circ$.

What you need: the distance from the centre to a chord is $\sqrt{R^2-\left(\frac{\text{chord}}{2}\right)^2}$.

Steps:

1. Find the distance from $O$ to each chord.
2. Look at two cases: the chords on the same side of $O$ or on opposite sides.