Practice: the circle, its equation, a line and a circle

Geometry, grade 11 (4 units) Lesson 5 of 20Practice

Practice: the circle, its equation, a line and a circle

$\displaystyle O$ is the centre of the circle, and $\displaystyle {\angle AOB}={96^{\circ}}$ .

How many degrees is the minor arc

\displaystyle AB

/ – 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

How many degrees is the major arc

\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

The centre of the circle is $\displaystyle (-2,5)$ , and the radius $\displaystyle 4$ .
Write the equation of the circle.

The circle: $\displaystyle {(x-7)^{2}+(y+3)^{2}}={50}$ .

What is the centre of the circle?

What is the radius? Leave the answer with a root, in its simplest form.

/ – 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 a circle: $\displaystyle A(-3,-2)$ and $\displaystyle B(5,4)$ .
Write the equation of the circle.

\displaystyle A=(-3,-2)

\displaystyle B=(5,4)

The radius of the circle is $\displaystyle 10$ , and the chord $\displaystyle AB$ is $\displaystyle 6$ from the centre.
How long is the chord?

/ – 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

The circle: $\displaystyle {x^{2}+y^{2}}={20}$ .
The point: $\displaystyle P(4,-3)$ .

Substitute

\displaystyle P

in the left side of the equation.

/ – 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

Is $\displaystyle P$ on the circle, inside it or outside it? Explain.

Find where the circle $\displaystyle {x^{2}+y^{2}}={25}$ and the line $\displaystyle {x+y}={7}$ meet.

The line $\displaystyle {y}={x+8}$ cuts the circle $\displaystyle {(x-2)^{2}+(y-5)^{2}}={25}$ at the points $\displaystyle A$ and $\displaystyle B$ .

Find the length of the chord

\displaystyle AB

. Leave the answer with a root, in its simplest form.

/ – 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

$\displaystyle O$ is the centre of the circle.
$\displaystyle AB$ and $\displaystyle AC$ are equal chords: $\displaystyle {AB}={AC}$ .
Prove: $\displaystyle {\angle BAO}={\angle CAO}$ .

$\displaystyle O$ is the centre of the circle.
$\displaystyle AB$ and $\displaystyle CD$ are parallel chords: $\displaystyle AB\parallel CD$ .
$\displaystyle M$ is the midpoint of the chord $\displaystyle AB$ .
Prove: $\displaystyle OM\perp CD$ .

Formulas: geometry, grade 11 (4 units)

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