Circles

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1. Inscribed and central angles

Warm-up

Three quick questions from earlier lessons.

Discover

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

  1. 50∘50^\circ
  2. 100∘100^\circ

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

  1. It stays the same: half of the central angle
  2. 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∘90^\circ.

Worked example

ABAB is a diameter, CC on the circle, ∠A=35∘\angle A = 35^\circ.

∠C=90∘\angle C = 90^\circ
Inscribed in a semicircle.
∠B=180−90−35=55∘\angle B = 180 - 90 - 35 = 55^\circ
Angle sum.

Your turn, guided

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

80∘80^\circ
Double.

Practice ★

  1. A central angle measures 49∘49^\circ. Find an inscribed angle that intercepts the same arc (degrees).
  2. A central angle measures 57∘57^\circ. Find an inscribed angle that intercepts the same arc (degrees).
  1. An inscribed angle measures 45∘45^\circ. Find the central angle on the same arc (degrees).

Watch out!

Practice ★★

  1. Triangle ABCABC is inscribed in a circle with ABAB a diameter, and ∠A=67∘\angle A = 67^\circ. Find ∠B\angle B (degrees).
  2. Triangle ABCABC is inscribed in a circle with ABAB a diameter, and ∠A=29∘\angle A = 29^\circ. Find ∠B\angle B (degrees).

Exit ticket

Three questions to check you have got it.

  1. A central angle measures 49∘49^\circ. Find an inscribed angle that intercepts the same arc (degrees).
  2. A central angle measures 57∘57^\circ. Find an inscribed angle that intercepts the same arc (degrees).
  3. A central angle measures 165∘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?

  1. 90∘90^\circ
  2. 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 PP 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.

1212
√(169 − 25).

Practice ★

  1. A circle has radius 6. Point PP is 10 from the center. Find the length of a tangent segment from PP to the circle.
  2. A circle has radius 3. Point PP is 5 from the center. Find the length of a tangent segment from PP to the circle.
  1. A line touches a circle of radius 7 at exactly one point TT. What is the angle between the line and the radius to TT?
    1. 90∘90^\circ
    2. It depends on the radius
    3. 0∘0^\circ: they lie along each other

Watch out!

Practice ★★

  1. Two tangents from PP touch a circle (center OO) at AA and BB, with ∠APB=61∘\angle APB = 61^\circ. Find ∠AOB\angle AOB (degrees).
  2. Two tangents from PP touch a circle (center OO) at AA and BB, with ∠APB=76∘\angle APB = 76^\circ. Find ∠AOB\angle AOB (degrees).

Exit ticket

Three questions to check you have got it.

  1. A circle has radius 6. Point PP is 10 from the center. Find the length of a tangent segment from PP to the circle.
  2. A circle has radius 3. Point PP is 5 from the center. Find the length of a tangent segment from PP to the circle.
  3. A circle has radius 12. Point PP is 20 from the center. Find the length of a tangent segment from PP 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?

  1. 180∘180^\circ
  2. 360∘360^\circ

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

  1. Always 180°
  2. Always 360°

Key idea

Cyclic quadrilaterals.

A quadrilateral inscribed in a circle has opposite angles supplementary. Conversely, if opposite angles add to 180∘180^\circ, the quadrilateral can be inscribed in a circle.

Worked example

∠A=75∘\angle A = 75^\circ, ∠B=110∘\angle B = 110^\circ.

∠C=105∘\angle C = 105^\circ
Opposite A.
∠D=70∘\angle D = 70^\circ
Opposite B.

Your turn, guided

Your turn: ∠P=98∘\angle P = 98^\circ in cyclic PQRSPQRS. Find ∠R\angle R.

82∘82^\circ
180 − 98.

Practice ★

  1. ABCDABCD is inscribed in a circle and ∠A=128∘\angle A = 128^\circ. Find ∠C\angle C (degrees).
  2. ABCDABCD is inscribed in a circle and ∠A=40∘\angle A = 40^\circ. Find ∠C\angle C (degrees).
  1. 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 ★★

  1. PQRSPQRS is inscribed in a circle with ∠Q=113∘\angle Q = 113^\circ. Find ∠S\angle S (degrees).
  2. PQRSPQRS is inscribed in a circle with ∠Q=42∘\angle Q = 42^\circ. Find ∠S\angle S (degrees).

Exit ticket

Three questions to check you have got it.

  1. ABCDABCD is inscribed in a circle and ∠A=128∘\angle A = 128^\circ. Find ∠C\angle C (degrees).
  2. ABCDABCD is inscribed in a circle and ∠A=40∘\angle A = 40^\circ. Find ∠C\angle C (degrees).
  3. ABCDABCD is inscribed in a circle and ∠A=107∘\angle A = 107^\circ. Find ∠C\angle C (degrees).

Summary card

Cyclic quadrilaterals

Opposite angles add to 180∘180^\circ.

Watch out: supplementary, not equal.

4. Arc length and sector area

Warm-up

Three quick questions from earlier lessons.

Discover

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

  1. 16\tfrac{1}{6}
  2. 160\tfrac{1}{60}

Key idea

Arcs and sectors are fractions of the circle.

For a central angle θ\theta in degrees: arc length =θ360⋅2πr= \dfrac{\theta}{360} \cdot 2\pi r; sector area =θ360⋅πr2= \dfrac{\theta}{360} \cdot \pi r^2.

Worked example

r=6r = 6, θ=120∘\theta = 120^\circ.

120360=13\tfrac{120}{360} = \tfrac{1}{3}
One third.
Arc =4π= 4\pi
⅓ · 12π.
Sector =12π= 12\pi
⅓ · 36π.

Your turn, guided

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

5π5\pi
¼ · 20π.

Practice ★

  1. A circle has radius 8 cm. Find the length of an arc with central angle 152∘152^\circ (exact, in terms of π).Not to scale
  2. A circle has radius 12 cm. Find the length of an arc with central angle 289∘289^\circ (exact, in terms of π).Not to scale
  1. Find the length of the arc cut off by a 90∘90^\circ central angle in a circle of radius 12 (exact).Not to scale

Watch out!

Practice ★★

  1. Find the area of a sector with radius 5 cm and central angle 188∘188^\circ (exact, in terms of π).Not to scale
  2. Find the area of a sector with radius 10 cm and central angle 147∘147^\circ (exact, in terms of π).Not to scale
  1. Boss problem. A slice of pizza is a sector of radius 6 cm with central angle 50∘50^\circ. Give exact answers in terms of π.

Exit ticket

Three questions to check you have got it.

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

Summary card

Arcs and sectors

Arc =θ3602πr= \tfrac{\theta}{360} 2\pi r, sector =θ360πr2= \tfrac{\theta}{360}\pi r^2.

Watch out: circumference for arcs, area for sectors.

5. Radian measure and arc length

Discover

Wrap a piece of string as long as the radius around the edge of a circle. The angle it covers at the center is the same for every circle. How many of these angles make a half turn?

  1. About 3.14, that is π\pi
  2. 180

Key idea

One radian is the angle that cuts off an arc as long as the radius.

180∘=π180^\circ = \pi radians. Degrees to radians: multiply by π180\dfrac{\pi}{180}. Radians to degrees: multiply by 180π\dfrac{180}{\pi}. Arc length: s=rθs = r\theta, with θ\theta in radians.

Worked example

Convert 135∘135^\circ to radians, then find the arc it cuts off on a circle of radius 4 cm.

135⋅π180=3π4135 \cdot \dfrac{\pi}{180} = \dfrac{3\pi}{4}
Multiply by π/180.
s=4⋅3π4=3πs = 4 \cdot \dfrac{3\pi}{4} = 3\pi cm
s = rθ, θ in radians.

Your turn, guided

Your turn: convert 5π6\dfrac{5\pi}{6} radians to degrees.

5⋅1806\dfrac{5 \cdot 180}{6}
Replace π by 180°.
=150∘= 150^\circ
900 ÷ 6.

Practice ★

  1. Convert 225∘225^\circ to radians. Give the exact value.
  2. Convert 330∘330^\circ to radians. Give the exact value.
  3. Convert 300∘300^\circ to radians. Give the exact value.
  4. Convert 30∘30^\circ to radians. Give the exact value.
  5. Convert 180∘180^\circ to radians. Give the exact value.

Watch out!

Practice ★★

  1. Convert π\pi radians to degrees.
  2. Convert 5π4\frac{5 \pi}{4} radians to degrees.
  3. Convert 3π3 \pi radians to degrees.
  1. A circle has radius 4 cm. What is the length of the arc cut off by a central angle of 3π4\frac{3 \pi}{4}? Give the exact value, in cm.
  2. A circle has radius 8 cm. What is the length of the arc cut off by a central angle of 2π3\frac{2 \pi}{3}? Give the exact value, in cm.
  3. A circle has radius 12 cm. What is the length of the arc cut off by a central angle of 5π6\frac{5 \pi}{6}? Give the exact value, in cm.

Exit ticket

Three questions to check you have got it.

  1. A circle has radius 4 cm. What is the length of the arc cut off by a central angle of 3π4\frac{3 \pi}{4}? Give the exact value, in cm.
  2. A circle has radius 8 cm. What is the length of the arc cut off by a central angle of 2π3\frac{2 \pi}{3}? Give the exact value, in cm.
  3. A circle has radius 12 cm. What is the length of the arc cut off by a central angle of 5π6\frac{5 \pi}{6}? Give the exact value, in cm.

Summary card

Radians

  • 180∘=π180^\circ = \pi rad.
  • Degrees → radians: ×π180\times \dfrac{\pi}{180}. Radians → degrees: ×180π\times \dfrac{180}{\pi}.
  • Arc length: s=rθs = r\theta, θ\theta in radians.

Watch out: never put degrees into s=rθs = r\theta.

Answers: Circles

1. Inscribed and central angles

A

  1. 492\frac{49}{2}
  2. 572\frac{57}{2}

B

  1. 9090

C

  1. 2323
  2. 6161

D

  1. 492\frac{49}{2}
  2. 572\frac{57}{2}
  3. 1652\frac{165}{2}

2. Tangent lines to a circle

A

  1. 88
  2. 44

B

  1. 90∘90^\circ

C

  1. 119119
  2. 104104

D

  1. 88
  2. 44
  3. 1616

3. Quadrilaterals inscribed in a circle

A

  1. 5252
  2. 140140

B

  1. A rectangle (for example 9 by 11)

C

  1. 6767
  2. 138138

D

  1. 5252
  2. 140140
  3. 7373

4. Arc length and sector area

A

  1. 304π45\frac{304 \pi}{45}
  2. 289π15\frac{289 \pi}{15}

B

  1. 6π6 \pi

C

  1. 235π18\frac{235 \pi}{18}
  2. 245π6\frac{245 \pi}{6}

D

  1. 5π5 \pi

E

  1. 304π45\frac{304 \pi}{45}
  2. 289π15\frac{289 \pi}{15}
  3. 91π15\frac{91 \pi}{15}

5. Radian measure and arc length

A

  1. 5π4\frac{5 \pi}{4}
  2. 11π6\frac{11 \pi}{6}
  3. 5π3\frac{5 \pi}{3}
  4. π6\frac{\pi}{6}
  5. π\pi

B

  1. 180180
  2. 225225
  3. 540540

C

  1. 3π3 \pi
  2. 16π3\frac{16 \pi}{3}
  3. 10π10 \pi

D

  1. 3π3 \pi
  2. 16π3\frac{16 \pi}{3}
  3. 10π10 \pi