Practice: angles in a circle

Geometry, grade 11 (4 units) Lesson 9 of 20Practice

Practice: angles in a circle

$\displaystyle O$ is the centre of the circle, and $\displaystyle {\angle AOB}={124^{\circ}}$ .
$\displaystyle C$ is on the major arc $\displaystyle AB$ , and $\displaystyle D$ on the minor arc $\displaystyle AB$ .

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

. (A hint: which arc does it stand on?)

/ – 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

$\displaystyle AB$ is a diameter of the circle, and $\displaystyle C$ is on the circle.
$\displaystyle {\angle ABC}={28^{\circ}}$ .
Find $\displaystyle \angle BAC$ .

/ – 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

In a circle of radius $\displaystyle 7$ cm, the central angle $\displaystyle \angle AOB$ is $\displaystyle 60^{\circ}$ .
How long is the chord $\displaystyle AB$ , in cm?

/ – 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

$\displaystyle A$ , $\displaystyle B$ , $\displaystyle C$ , $\displaystyle D$ are on a circle, and the chords $\displaystyle AC$ and $\displaystyle BD$ meet at $\displaystyle E$ .
$\displaystyle {\angle BAC}={32^{\circ}}$ and $\displaystyle {\angle ABD}={48^{\circ}}$ .

Find

\displaystyle \angle BDC

/ – 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 ACD

/ – 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 AEB

/ – 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

$\displaystyle AB$ is a diameter of the circle with centre $\displaystyle O$ .
The points $\displaystyle C$ and $\displaystyle D$ are on the circle, on the two sides of $\displaystyle AB$ .
Prove: $\displaystyle {\angle ACB}={\angle ADB}$ .

The quadrilateral $\displaystyle ABCD$ is inscribed in a circle, and $\displaystyle {AB}={CD}$ .
Prove: $\displaystyle {\angle ACB}={\angle CBD}$ .

The triangle $\displaystyle ABC$ is inscribed in a circle with centre $\displaystyle O$ .
$\displaystyle OM\perp AB$ , and $\displaystyle M$ is on $\displaystyle AB$ .
Prove: $\displaystyle {\angle AOM}={\angle ACB}$ .

The triangle $\displaystyle ABC$ is inscribed in a circle with centre $\displaystyle O$ , as in question 7.
The distance of the centre from the side $\displaystyle AB$ is $\displaystyle {OM}={8}$ , and $\displaystyle {\angle ACB}={72^{\circ}}$ .
Round to two decimal places.

How many degrees is

\displaystyle \angle AOM

/ – 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 radius of the circle.

/ – 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 side

\displaystyle AB

/ – 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

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
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi R^{2}$