Constructions: Tangents and Circles — chapter notes

• A tangent meets a circle in one point, the point of contact $T$.
• The radius $OT$ is perpendicular to the tangent; every other point of the tangent is outside the circle.
• From a point on the circle there is one tangent; from a point inside there is none; from a point outside there are two.

Example: $OP = 5$, $r = 3$: $PT = \sqrt{5^2 - 3^2} = 4$.

• Steps of construction: join $OP$; bisect it at $M$; with centre $M$ and radius $MO$ draw a circle; it cuts the given circle at $A$ and $B$; join $PA$ and $PB$.
• Why it works: $A$ is on the circle with diameter $OP$, so $\angle OAP = 90^\circ$ (angle in a semicircle).
• The tangents are equal, $PA = PB$, and $PO$ bisects $\angle APB$.
• $\sin\angle APO = \dfrac{r}{OP}$, so $\angle APB = 2\sin^{-1}\dfrac{r}{OP}$ and $\angle AOB = 180^\circ - \angle APB$.

Example: $r = 3$, $OP = 5$: $PA = 4$, $\angle APB \approx 74^\circ$, $\angle AOB \approx 106^\circ$.

• A chord of the larger circle (radius $R$) that touches the smaller one (radius $r$) is bisected at the point of contact.
• Its length is $2\sqrt{R^2 - r^2}$.
• Tangents from a point of the outer circle to the inner circle are parts of such chords.

Example: $R = 5$, $r = 3$: chord $= 2 \times 4 = 8$.

• The perpendicular bisectors of the sides meet at the circumcentre $O$; $OA = OB = OC = R$.
• Steps: draw the triangle; perpendicular bisectors of two sides meet at $O$; circle with centre $O$ and radius $OA$; measure $R$.
• Acute triangle: $O$ inside. Right triangle: $O$ is the mid-point of the hypotenuse, $R = \dfrac{c}{2}$. Obtuse triangle: $O$ outside.
• Check: $R = \dfrac{abc}{4 \times \text{area}}$.

Example: $BC = 7$, $AB = 5$, $\angle B = 60^\circ$: $R \approx 3.6$ cm.

• The angle bisectors meet at the incentre $I$, equally far from the three sides; $ID = IE = IF = r$, each perpendicular to its side.
• Steps: draw the triangle; bisect two angles; they meet at $I$; draw $ID$ perpendicular to a side; circle with centre $I$ and radius $ID$.
• $r = \dfrac{\text{area}}{s}$ where $s$ is the semi-perimeter; for a right triangle $r = \dfrac{a + b - c}{2}$.
• Tangents from a vertex are equal: $AF = AE = s - a$.

Example: sides $6.5$, $7$, $7.5$: $s = 10.5$, area $21$, $r = 2$ cm.

• A regular hexagon is six equilateral triangles around the centre $O$: side $a$, $R = a$, each angle at $O$ is $60^\circ$.
• Steps: circle of radius $a$ about $O$; mark $A$; with the same radius cut $B$, $C$, $D$, $E$, $F$ round the circle; join in order.
• The circle drawn is the circumcircle. The incircle has centre $O$ and radius $OM$ with $M$ the mid-point of a side: $r = a\sin 60^\circ = \dfrac{a\sqrt{3}}{2}$.
• Perimeter $6a$.

Example: $a = 4$: $R = 4$, $r = \sqrt{12} \approx 3.5$.

• Measure lengths to 0.1 cm and angles to the nearest degree; write the unit.
• Always check a measured value with a calculation: $\sqrt{OP^2 - r^2}$ for tangents; $\dfrac{abc}{4K}$ and $\dfrac{K}{s}$ for the two radii; $a\sin 60^\circ$ for the hexagon.
• Keep every construction arc; label every point.
• Ring between two circles: $\pi(R^2 - r^2)$.

Example: hexagon side $5$: ring area $3.14 \times 6.25 \approx 19.6$ cm$^2$.