A quadrilateral inscribed in a circle

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

A quadrilateral inscribed in a circle

The opposite angles of an inscribed quadrilateral

A quadrilateral is inscribed in a circle when its four vertices are on the circle.

Theorem: A quadrilateral can be inscribed in a circle if and only if a pair of its opposite angles adds up to $\displaystyle 180^{\circ}$ .

\displaystyle \angle A+\angle C

\displaystyle =

\displaystyle 180^{\circ}

We prove the first direction.
Given: the quadrilateral $\displaystyle ABCD$ is inscribed in a circle.
To prove: $\displaystyle {\angle BAD+\angle BCD}={180^{\circ}}$ .

The proof below draws the diagonals, and replaces each of the two parts of $\displaystyle \angle BAD$ by an equal inscribed angle in the triangle $\displaystyle BCD$ :

	Claim	Reason
1	$\displaystyle {\angle BAC}={\angle BDC}$	Inscribed angles on the same arc $\displaystyle BC$
2	$\displaystyle {\angle CAD}={\angle CBD}$	Inscribed angles on the same arc $\displaystyle CD$
3	$\displaystyle {\angle BAD}={\angle BDC+\angle CBD}$	(1), (2); $\displaystyle {\angle BAD}={\angle BAC+\angle CAD}$
4	$\displaystyle {\angle BCD+\angle CBD+\angle BDC}={180^{\circ}}$	The angles of the triangle $\displaystyle BCD$
5	$\displaystyle {\angle BAD+\angle BCD}={180^{\circ}}$	(3), (4)

The other direction

If a pair of opposite angles of a quadrilateral adds up to $\displaystyle 180^{\circ}$ , then it can be inscribed in a circle.

The other pair of opposite angles then adds up to $\displaystyle 180^{\circ}$ too: the angles of a quadrilateral add up to $\displaystyle 360^{\circ}$ .
A rectangle and a square can be inscribed in a circle: all their angles are $\displaystyle 90^{\circ}$ .
A parallelogram that is not a rectangle cannot: its opposite angles are equal, so they add up to $\displaystyle 180^{\circ}$ only when each is $\displaystyle 90^{\circ}$ .

Exercise 1

The quadrilateral $\displaystyle ABCD$ is inscribed in a circle.
$\displaystyle {\angle A}={75^{\circ}}$ and $\displaystyle {\angle B}={110^{\circ}}$ .

Find

\displaystyle \angle C

/ – 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 D

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Exercise 2

The quadrilateral $\displaystyle ABCD$ is inscribed in a circle.
The opposite angles $\displaystyle \angle A$ and $\displaystyle \angle C$ are in the ratio $\displaystyle 2:3$ .

Find

\displaystyle \angle A

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Find

\displaystyle \angle C

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Exercise 3

A hint: what is

\displaystyle \angle D+\angle B

in the inscribed quadrilateral? And

\displaystyle \angle C+\angle B

, beside the trapezoid’s leg

\displaystyle BC

Theorem: If a trapezoid is inscribed in a circle, then it is an isosceles trapezoid.

Prove the theorem yourself.
Given: the trapezoid $\displaystyle ABCD$ is inscribed in a circle, $\displaystyle AB\parallel DC$ .
To prove: $\displaystyle {AD}={BC}$ .

Exercise 4

The quadrilateral $\displaystyle ABCD$ is inscribed in the circle $\displaystyle {(x-7)^{2}+(y-20)^{2}}={169}$ .
Its vertices are $\displaystyle A(-6,20)$ , $\displaystyle B(2,32)$ , $\displaystyle C(12,32)$ and $\displaystyle D(20,20)$ .

\displaystyle (x-7)^{2}+(y-20)^{2}=169

\displaystyle A=(-6,20)

\displaystyle B=(2,32)

\displaystyle C=(12,32)

\displaystyle D=(20,20)

Substitute

\displaystyle B

in the left side of the circle’s equation. Is

\displaystyle B

on the circle?

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Explain why $\displaystyle AD\parallel BC$ .

Find the length of the leg

\displaystyle AB

. (

\displaystyle CD

is found the same way, and has the same length: the trapezoid is isosceles.)

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Find

\displaystyle \angle DAB

from the slope of

\displaystyle AB

, rounded to two decimal places.

/ – 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 ABC

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Find the area of the trapezoid.

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