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1. Inscribed and central angles

Warm-up

Three quick questions from earlier lessons.

Discover

A central angle of $100^\circ$ and an inscribed angle stand on the same arc. Which is the inscribed angle?

$50^\circ$
$100^\circ$

Slide C along the arc. Watch the angle at C.

It stays the same: half of the central angle
It gets bigger as C rises

Key idea

Angles in a circle depend on the arc they stand on.

An inscribed angle is half the central angle that intercepts the same arc. Inscribed angles on the same arc are equal. An angle inscribed in a semicircle (standing on a diameter) is

90^\circ

Worked example

$AB$ is a diameter, $C$ on the circle, $\angle A = 35^\circ$ .

\angle C = 90^\circ

Inscribed in a semicircle.

\angle B = 180 - 90 - 35 = 55^\circ

Angle sum.

Your turn, guided

Your turn: an inscribed angle of $40^\circ$ . The central angle on the same arc.

80^\circ

Double.

Practice ★

A central angle measures $49^\circ$ . Find an inscribed angle that intercepts the same arc (degrees).
A central angle measures $57^\circ$ . Find an inscribed angle that intercepts the same arc (degrees).

An inscribed angle measures $45^\circ$ . Find the central angle on the same arc (degrees).

Watch out!

Practice ★★

Triangle $ABC$ is inscribed in a circle with $AB$ a diameter, and $\angle A = 67^\circ$ . Find $\angle B$ (degrees).
Triangle $ABC$ is inscribed in a circle with $AB$ a diameter, and $\angle A = 29^\circ$ . Find $\angle B$ (degrees).

Exit ticket

Three questions to check you have got it.

A central angle measures $49^\circ$ . Find an inscribed angle that intercepts the same arc (degrees).
A central angle measures $57^\circ$ . Find an inscribed angle that intercepts the same arc (degrees).
A central angle measures $165^\circ$ . Find an inscribed angle that intercepts the same arc (degrees).

Summary card

Inscribed angles

Inscribed = ½ central; on a diameter: 90°.

Watch out: halve, do not double.

2. Tangent lines to a circle

Warm-up

Three quick questions from earlier lessons.

Discover

A wheel rolls on the ground. The ground touches it at one point. What angle does the spoke to that point make with the ground?

$90^\circ$
It depends on the speed

Key idea

Tangents and radii are perpendicular.

A tangent touches the circle at one point and is perpendicular to the radius there. The two tangent segments from an outside point are equal in length.

Worked example

Radius 6, point $P$ at distance 10 from the center.

Right triangle with hypotenuse 10, leg 6

Right angle at the point of contact.

Tangent length 8

√(100 − 36).

Your turn, guided

Your turn: radius 5, distance 13. The tangent length.

12

√(169 − 25).

Practice ★

A circle has radius 6. Point $P$ is 10 from the center. Find the length of a tangent segment from $P$ to the circle.
A circle has radius 3. Point $P$ is 5 from the center. Find the length of a tangent segment from $P$ to the circle.

A line touches a circle of radius 7 at exactly one point $T$ . What is the angle between the line and the radius to $T$ ?
1. $90^\circ$
2. It depends on the radius
3. $0^\circ$ : they lie along each other

Watch out!

Practice ★★

Two tangents from $P$ touch a circle (center $O$ ) at $A$ and $B$ , with $\angle APB = 61^\circ$ . Find $\angle AOB$ (degrees).
Two tangents from $P$ touch a circle (center $O$ ) at $A$ and $B$ , with $\angle APB = 76^\circ$ . Find $\angle AOB$ (degrees).

Exit ticket

Three questions to check you have got it.

A circle has radius 6. Point $P$ is 10 from the center. Find the length of a tangent segment from $P$ to the circle.
A circle has radius 3. Point $P$ is 5 from the center. Find the length of a tangent segment from $P$ to the circle.
A circle has radius 12. Point $P$ is 20 from the center. Find the length of a tangent segment from $P$ to the circle.

Summary card

Tangent lines

Tangent ⊥ radius; tangent segments from a point are equal.

Watch out: a right triangle, not a subtraction.

3. Quadrilaterals inscribed in a circle

Warm-up

Three quick questions from earlier lessons.

Discover

A quadrilateral has all four vertices on a circle. Opposite angles stand on arcs that together make the whole circle. What do opposite angles add to?

$180^\circ$
$360^\circ$

Move D around the circle. Add the angles at A and C.

Always 180°
Always 360°

Key idea

Cyclic quadrilaterals.

A quadrilateral inscribed in a circle has opposite angles supplementary. Conversely, if opposite angles add to

180^\circ

, the quadrilateral can be inscribed in a circle.

Worked example

$\angle A = 75^\circ$ , $\angle B = 110^\circ$ .

\angle C = 105^\circ

Opposite A.

\angle D = 70^\circ

Opposite B.

Your turn, guided

Your turn: $\angle P = 98^\circ$ in cyclic $PQRS$ . Find $\angle R$ .

82^\circ

180 − 98.

Practice ★

$ABCD$ is inscribed in a circle and $\angle A = 128^\circ$ . Find $\angle C$ (degrees).
$ABCD$ is inscribed in a circle and $\angle A = 40^\circ$ . Find $\angle C$ (degrees).

Which quadrilateral can always be inscribed in a circle?
1. A rectangle (for example 9 by 11)
2. Any parallelogram
3. Any rhombus

Watch out!

Practice ★★

$PQRS$ is inscribed in a circle with $\angle Q = 113^\circ$ . Find $\angle S$ (degrees).
$PQRS$ is inscribed in a circle with $\angle Q = 42^\circ$ . Find $\angle S$ (degrees).

Exit ticket

Three questions to check you have got it.

$ABCD$ is inscribed in a circle and $\angle A = 128^\circ$ . Find $\angle C$ (degrees).
$ABCD$ is inscribed in a circle and $\angle A = 40^\circ$ . Find $\angle C$ (degrees).
$ABCD$ is inscribed in a circle and $\angle A = 107^\circ$ . Find $\angle C$ (degrees).

Summary card

Cyclic quadrilaterals

Opposite angles add to $180^\circ$ .

Watch out: supplementary, not equal.

4. Arc length and sector area

Warm-up

Three quick questions from earlier lessons.

Discover

A $60^\circ$ slice is cut from a round pie. What fraction of the whole pie is it?

$\tfrac{1}{6}$
$\tfrac{1}{60}$

Key idea

Arcs and sectors are fractions of the circle.

For a central angle

\theta

in degrees: arc length

= \dfrac{\theta}{360} \cdot 2\pi r

; sector area

= \dfrac{\theta}{360} \cdot \pi r^2

Worked example

$r = 6$ , $\theta = 120^\circ$ .

\tfrac{120}{360} = \tfrac{1}{3}

One third.

Arc

= 4\pi

⅓ · 12π.

Sector

= 12\pi

⅓ · 36π.

Your turn, guided

Your turn: $r = 10$ , $\theta = 90^\circ$ . The arc length (in terms of π).

5\pi

¼ · 20π.

Practice ★

A circle has radius 8 cm. Find the length of an arc with central angle $152^\circ$ (exact, in terms of π).Not to scale
A circle has radius 12 cm. Find the length of an arc with central angle $289^\circ$ (exact, in terms of π).Not to scale

Find the length of the arc cut off by a $90^\circ$ central angle in a circle of radius 12 (exact).Not to scale

Watch out!

Practice ★★

Find the area of a sector with radius 5 cm and central angle $188^\circ$ (exact, in terms of π).Not to scale
Find the area of a sector with radius 10 cm and central angle $147^\circ$ (exact, in terms of π).Not to scale

Boss problem. A slice of pizza is a sector of radius 6 cm with central angle $50^\circ$ . Give exact answers in terms of π.

Exit ticket

Three questions to check you have got it.

A circle has radius 8 cm. Find the length of an arc with central angle $152^\circ$ (exact, in terms of π).Not to scale
A circle has radius 12 cm. Find the length of an arc with central angle $289^\circ$ (exact, in terms of π).Not to scale
A circle has radius 12 cm. Find the length of an arc with central angle $91^\circ$ (exact, in terms of π).Not to scale

Summary card

Arcs and sectors

Arc $= \tfrac{\theta}{360} 2\pi r$ , sector $= \tfrac{\theta}{360}\pi r^2$ .

Watch out: circumference for arcs, area for sectors.

5. Circles: equation, center and radius

Warm-up

Three quick questions from earlier lessons.

Discover

A point $(x, y)$ is on the circle with center $(2, 1)$ and radius 3 when its distance to $(2, 1)$ is 3. Which equation says that?

$(x - 2)^2 + (y - 1)^2 = 9$
$(x + 2)^2 + (y + 1)^2 = 9$
$(x - 2)^2 + (y - 1)^2 = 3$

Key idea

A circle is the set of points at a fixed distance $r$ from a center $(h, k)$ .

(x - h)^2 + (y - k)^2 = r^2

. From the expanded form

x^2 + y^2 + Dx + Ey + F = 0

, complete the square in

x

and in

y

to read the center and radius.

Worked example

Find the center and radius of $x^2 + y^2 - 4x + 6y - 3 = 0$ .

(x - 2)^2 - 4 + (y + 3)^2 - 9 - 3 = 0

Group and complete the squares: x² − 4x = (x − 2)² − 4, y² + 6y = (y + 3)² − 9.

(x - 2)^2 + (y + 3)^2 = 16

Move the numbers to the right.

Center

(2, -3)

, radius

4

Read h, k and r = √16.

Your turn, guided

Your turn: $(x + 5)^2 + (y - 1)^2 = 49$ .

Center

(-5, 1)

x + 5 = x − (−5).

Radius

7

√49.

Practice ★

What is the radius of the circle $(x - 9)^2 + (y + 1)^2 = 16$ ?
What is the radius of the circle $(x - 1)^2 + (y + 7)^2 = 64$ ?
What is the radius of the circle $(x - 4)^2 + (y + 9)^2 = 16$ ?

Watch out!

Practice ★★

Find the center and radius of the circle $x^2 + y^2 - 10x + 4y + 25 = 0$ .
Find the center and radius of the circle $x^2 + y^2 - 2x + 4y + -11 = 0$ .

Which is the equation of the circle with center $(2, -8)$ and radius 7?
1. $(x - 2)^2 + (y + 8)^2 = 49$
2. $(x + 2)^2 + (y - 8)^2 = 49$
3. $(x - 2)^2 + (y + 8)^2 = 7$
Which is the equation of the circle with center $(7, -2)$ and radius 9?
1. $(x - 7)^2 + (y + 2)^2 = 81$
2. $(x + 7)^2 + (y - 2)^2 = 81$
3. $(x - 7)^2 + (y + 2)^2 = 9$

Exit ticket

Three questions to check you have got it.

Which is the equation of the circle with center $(2, -8)$ and radius 7?
1. $(x - 2)^2 + (y + 8)^2 = 49$
2. $(x + 2)^2 + (y - 8)^2 = 49$
3. $(x - 2)^2 + (y + 8)^2 = 7$
Which is the equation of the circle with center $(7, -2)$ and radius 9?
1. $(x - 7)^2 + (y + 2)^2 = 81$
2. $(x + 7)^2 + (y - 2)^2 = 81$
3. $(x - 7)^2 + (y + 2)^2 = 9$
Which is the equation of the circle with center $(6, -1)$ and radius 6?
1. $(x - 6)^2 + (y + 1)^2 = 36$
2. $(x + 6)^2 + (y - 1)^2 = 36$
3. $(x - 6)^2 + (y + 1)^2 = 6$

Summary card

Circles

$(x - h)^2 + (y - k)^2 = r^2$ : center $(h, k)$ , radius $r$ .
Expanded form: complete the square in $x$ and in $y$ .

Watch out: $(x + 3)$ means $h = -3$ , and the right side is $r^2$ .

6. Parabolas: focus and directrix

Warm-up

Three quick questions from earlier lessons.

Discover

A satellite dish is a parabola: every signal coming straight in bounces to one point, the focus. For $x^2 = 8y$ , where is it?

$(0, 2)$
$(0, 8)$

Key idea

A parabola is the set of points as far from a point (the focus) as from a line (the directrix).

x^2 = 4py

: vertex

(0, 0)

, focus

(0, p)

, directrix

y = -p

, opens up if

p > 0

y^2 = 4px

: focus

(p, 0)

, directrix

x = -p

, opens right if

p > 0

Worked example

Find the focus and directrix of $y^2 = -12x$ .

4p = -12

, so

p = -3

Match with y² = 4px.

Focus

(-3, 0)

: it opens to the left

Focus (p, 0).

Directrix

x = 3

Directrix x = −p.

Your turn, guided

Your turn: $x^2 = 20y$ .

p = 5

, focus

(0, 5)

4p = 20.

Directrix

y = -5

y = −p: type the number.

Practice ★

The parabola $x^2 = 8y$ has its focus at $(0, c)$ . What is $c$ ?
The parabola $x^2 = 24y$ has its focus at $(0, c)$ . What is $c$ ?
The parabola $x^2 = 4y$ has its focus at $(0, c)$ . What is $c$ ?

Watch out!

Practice ★★

The parabola $y^2 = 8x$ has the directrix $x = d$ . What is $d$ ?
The parabola $y^2 = 20x$ has the directrix $x = d$ . What is $d$ ?
The parabola $y^2 = 4x$ has the directrix $x = d$ . What is $d$ ?

A parabola has focus $(0, 3)$ and directrix $y = -3$ . Its equation is $x^2 = ky$ . What is $k$ ?
A parabola has focus $(0, 2)$ and directrix $y = -2$ . Its equation is $x^2 = ky$ . What is $k$ ?

Exit ticket

Three questions to check you have got it.

The parabola $y^2 = 8x$ has the directrix $x = d$ . What is $d$ ?
The parabola $y^2 = 20x$ has the directrix $x = d$ . What is $d$ ?
The parabola $y^2 = 4x$ has the directrix $x = d$ . What is $d$ ?

Summary card

Parabolas

$x^2 = 4py$ : focus $(0, p)$ , directrix $y = -p$ .
$y^2 = 4px$ : focus $(p, 0)$ , directrix $x = -p$ .

Watch out: divide by 4 to get $p$ .

7. Partitioning a segment

Warm-up

Three quick questions from earlier lessons.

Discover

From $A(0, 0)$ to $B(6, 3)$ . Where is the point one third of the way along?

$(2, 1)$
$(3, 1.5)$

Key idea

Move a fraction of the way, or take a weighted average.

The point a fraction

t

of the way from

A

B

A + t(B - A)

. A point dividing

AB

in the ratio

m : n

(from

A

) is

\left(\dfrac{n x_A + m x_B}{m + n}, \dfrac{n y_A + m y_B}{m + n}\right)

, a weighted average.

Worked example

$A(1, 2)$ , $B(9, 6)$ , ratio $1 : 3$ .

t = \tfrac{1}{4}

1 part of 4.

P = (3, 3)

¼ of (8, 4) = (2, 1).

Your turn, guided

Your turn: x-coordinate halfway between 2 and 10.

6

Average.

Practice ★

Tap the point one third of the way from $A(-3, -2)$ to $B(3, 4)$ .
Tap the point one third of the way from $A(-5, -6)$ to $B(4, 3)$ .

Point $P$ divides segment $AB$ in the ratio $1 : 2$ (closer to $A$ ). $A$ has $x = -4$ and $B$ has $x = 2$ . Find the $x$ -coordinate of $P$ .

Watch out!

Practice ★★

Point $P$ divides $AB$ in the ratio $3 : 1$ from $A$ . $A$ has $x = -2$ , $B$ has $x = 1$ . Find $P$ 's $x$ -coordinate.
Point $P$ divides $AB$ in the ratio $1 : 4$ from $A$ . $A$ has $x = -2$ , $B$ has $x = 2$ . Find $P$ 's $x$ -coordinate.

Exit ticket

Three questions to check you have got it.

Tap the point one third of the way from $A(-3, -2)$ to $B(3, 4)$ .
Tap the point one third of the way from $A(-5, -6)$ to $B(4, 3)$ .
Tap the point one third of the way from $A(-7, -2)$ to $B(2, 1)$ .

Summary card

Partitioning a segment

$A + t(B - A)$ ; ratio $m : n$ gives $t = \dfrac{m}{m + n}$ .

Watch out: count the parts of the ratio.

Answers: Circles with and without coordinates

1. Inscribed and central angles

$\frac{49}{2}$
$\frac{57}{2}$

$90$

$23$
$61$

$\frac{49}{2}$
$\frac{57}{2}$
$\frac{165}{2}$

2. Tangent lines to a circle

$90^\circ$

$119$
$104$

$8$
$4$
$16$

3. Quadrilaterals inscribed in a circle

$52$
$140$

A rectangle (for example 9 by 11)

$67$
$138$

$52$
$140$
$73$

4. Arc length and sector area

$\frac{304 \pi}{45}$
$\frac{289 \pi}{15}$

$6 \pi$

$\frac{235 \pi}{18}$
$\frac{245 \pi}{6}$

$5 \pi$

$\frac{304 \pi}{45}$
$\frac{289 \pi}{15}$
$\frac{91 \pi}{15}$

5. Circles: equation, center and radius

$(x - 2)^2 + (y + 8)^2 = 49$
$(x - 7)^2 + (y + 2)^2 = 81$

$(x - 2)^2 + (y + 8)^2 = 49$
$(x - 7)^2 + (y + 2)^2 = 81$
$(x - 6)^2 + (y + 1)^2 = 36$

6. Parabolas: focus and directrix

$-2$
$-5$
$-1$

$12$
$8$

$-2$
$-5$
$-1$

7. Partitioning a segment

$(-1, 0)$
$(-2, -3)$

$-2$

$\frac{1}{4}$
$- \frac{6}{5}$

$(-1, 0)$
$(-2, -3)$
$(-4, -1)$