The inscribed angle

Geometry, grade 11 (4 units) Lesson 7 of 20Learning

The inscribed angle

An inscribed angle is an angle whose vertex is on the circle and whose two arms are chords.

In the drawing, $\displaystyle \angle ACB$ is an inscribed angle: its vertex $\displaystyle C$ is on the circle, and $\displaystyle CA$ , $\displaystyle CB$ are chords.
It stands on the arc $\displaystyle AB$ – the arc inside the angle, the one without $\displaystyle C$ on it.
The central angle $\displaystyle \angle AOB$ stands on the same arc.

The inscribed angle and the central angle

Here the centre

\displaystyle O

is inside the angle

\displaystyle ACB

. When the centre is on one of its arms or outside it, the proof is much the same: the same diameter

\displaystyle CD

, and in the last step we subtract instead of adding.

Theorem: In a circle, an inscribed angle is half the central angle standing on the same arc.

Given: $\displaystyle O$ is the centre of the circle; $\displaystyle A$ , $\displaystyle B$ , $\displaystyle C$ are on the circle.
To prove: $\displaystyle {\angle AOB}={2\angle ACB}$ .

The proof below draws the diameter $\displaystyle CD$ and shows that each central angle on one side of it is twice the inscribed angle beside it – by an isosceles triangle and an exterior angle:

	Claim	Reason
1	$\displaystyle {OA}={OC}$	Radii of the circle
2	$\displaystyle {\angle OAC}={\angle OCA}$	(1); the base angles of an isosceles triangle are equal
3	$\displaystyle {\angle AOD}={\angle OAC+\angle OCA}$	An exterior angle of the triangle $\displaystyle AOC$ is the sum of the two interior angles not next to it
4	$\displaystyle {\angle AOD}={2\angle ACO}$	(2), (3)
5	$\displaystyle {\angle BOD}={2\angle BCO}$	In the same way, in the isosceles triangle $\displaystyle BOC$ ( $\displaystyle {OB}={OC}$ )
6	$\displaystyle {\angle AOB}={2\angle ACB}$	(4), (5); we add the two, since $\displaystyle {\angle AOB}={\angle AOD+\angle BOD}$ and $\displaystyle {\angle ACB}={\angle ACO+\angle BCO}$

The inscribed angle and its arc

The measure of an arc is the central angle standing on it. So the theorem can also be said this way: an inscribed angle is half the arc it stands on.

\displaystyle \angle ACB

\displaystyle =

\displaystyle \frac{1}{2}\angle AOB

Example: the arc $\displaystyle AB$ is $\displaystyle 100^{\circ}$ . Every inscribed angle standing on it is $\displaystyle {\frac{100^{\circ}}{2}}={50^{\circ}}$ .

Inscribed angles on the same arc

Theorem: In a circle, inscribed angles standing on the same arc are equal.

Given: $\displaystyle C$ and $\displaystyle D$ are on the circle, on the same side of the chord $\displaystyle AB$ .
To prove: $\displaystyle {\angle ACB}={\angle ADB}$ .

The proof below compares each of the angles with the same central angle, $\displaystyle \angle AOB$ :

	Claim	Reason
1	$\displaystyle {\angle ACB}={\frac{1}{2}\angle AOB}$	An inscribed angle is half the central angle on the same arc
2	$\displaystyle {\angle ADB}={\frac{1}{2}\angle AOB}$	The same theorem
3	$\displaystyle {\angle ACB}={\angle ADB}$	(1), (2)

Equal arcs, chords and inscribed angles

For the same reason – every inscribed angle is half its arc – these theorems follow too:

In a circle, equal inscribed angles stand on equal arcs and equal chords.
In a circle, equal arcs have equal inscribed angles.

They hold in two equal circles (with equal radii) as well.

Exercise 1

$\displaystyle O$ is the centre of the circle, and $\displaystyle {\angle AOB}={140^{\circ}}$ .
$\displaystyle C$ and $\displaystyle D$ are on the circle, on the same side of $\displaystyle AB$ .

Find

\displaystyle \angle ACB

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Find

\displaystyle \angle ADB

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

The triangle $\displaystyle ABC$ is inscribed in a circle with centre $\displaystyle O$ and radius $\displaystyle 6$ .
$\displaystyle {\angle BAC}={30^{\circ}}$ .

Find

\displaystyle \angle BOC

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Find the length of the chord

\displaystyle BC

. (A hint: what kind of triangle is

\displaystyle BOC

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

$\displaystyle A$ , $\displaystyle B$ , $\displaystyle C$ , $\displaystyle D$ are on a circle.
The chords $\displaystyle AC$ and $\displaystyle BD$ meet at $\displaystyle E$ .
Prove: $\displaystyle \triangle AED\sim\triangle BEC$ .

Exercise 4

The quadrilateral $\displaystyle ABCD$ is inscribed in a circle.
$\displaystyle {\angle BAC}={\angle CAD}$ .
Prove: $\displaystyle {BC}={CD}$ .

Formulas: geometry, grade 11 (4 units)

Open as a page

The equation of a circle

The canonical equation – the centre at the origin, the radius $\displaystyle R$
$\displaystyle x^{2}+y^{2}=R^{2}$
The general equation – the centre $\displaystyle (a,b)$ , the radius $\displaystyle R$
$\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}$
The distance between two points
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}$
The midpoint of a segment
$\displaystyle x_M=\frac{x_1+x_2}{2},\ y_M=\frac{y_1+y_2}{2}$
Perpendicular lines – $\displaystyle m_1$ , $\displaystyle m_2$ their slopes
$\displaystyle m_1\cdot m_2$ $\displaystyle =$ $\displaystyle -1$

Angles and chords in a circle

An inscribed angle – half the central angle on the same arc
$\displaystyle \angle ACB$ $\displaystyle =$ $\displaystyle \frac{1}{2}\angle AOB$
A quadrilateral inscribed in a circle – a pair of opposite angles
$\displaystyle \angle A+\angle C$ $\displaystyle =$ $\displaystyle 180^{\circ}$
A chord $\displaystyle AB$ – $\displaystyle d$ its distance from the centre
$\displaystyle \left(\frac{AB}{2}\right)^{2}+d^{2}$ $\displaystyle =$ $\displaystyle R^{2}$

Tangents

Two tangents from a point $\displaystyle P$ – $\displaystyle A$ , $\displaystyle B$ the points of contact
$\displaystyle PA=PB$
The tangent segment $\displaystyle PA$ – $\displaystyle O$ the centre
$\displaystyle PA^{2}+R^{2}$ $\displaystyle =$ $\displaystyle PO^{2}$

Trigonometry, circumference and area

The law of sines – $\displaystyle R$ the radius of the circle through the triangle’s vertices
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The area of a triangle – $\displaystyle \gamma$ the angle between the sides $\displaystyle a$ and $\displaystyle b$
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{1}{2}ab\sin\gamma$
The area of a parallelogram – $\displaystyle a$ , $\displaystyle b$ adjacent sides, $\displaystyle \alpha$ the angle between them
$\displaystyle A$ $\displaystyle =$ $\displaystyle ab\sin\alpha$
The sine of an obtuse angle
$\displaystyle \sin(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The circumference
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi R$
The area of a disc
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi R^{2}$