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Naming the parts of a circle, the inscribed angle and tangent theorems, and arcs, sectors and radian measure.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you name the parts of a circle on a diagram, and then prove and use the theorems that follow from one fact: two radii of a circle make an isosceles triangle. You find inscribed angles from their arcs, use the right angle a tangent makes with the radius at its point of contact, and use the opposite angles of a cyclic quadrilateral. Then you treat an arc and a sector as fractions of a whole circle, and meet the radian, which measures an angle by how many radiuses fit along its arc.
You know that the angles of a triangle add to $180^\circ$, that the two base angles of an isosceles triangle are equal, and that every radius of one circle is the same length. That last fact is doing almost all the work below: any triangle with two radii for sides is isosceles, and that is where every one of these theorems comes from.
Centre: the point every point of the circle is the same distance from. Radius: a segment from the centre to the circle; diameter: a chord through the centre, twice a radius.
Chord: a segment joining two points on the circle. Arc: a piece of the circle itself, measured in degrees by the angle it makes at the centre.
Tangent: a line meeting the circle at exactly one point, the point of contact.
Inscribed angle: an angle whose vertex is on the circle and whose arms are chords. It subtends, or stands on, the arc between those chords.
Cyclic quadrilateral: a quadrilateral with all four corners on one circle.
An inscribed angle is half the arc it stands on, so an arc of $100^\circ$ gives an inscribed angle of $50^\circ$; the angle at the centre gets the whole $100^\circ$. Because a diameter cuts off an arc of $180^\circ$, the angle in a semicircle is a right angle. A tangent is perpendicular to the radius at its point of contact, and the two tangents from one external point are equal, so if one is $8$ cm the other is $8$ cm. The opposite angles of a cyclic quadrilateral add to $180^\circ$: an angle of $110^\circ$ faces one of $70^\circ$, because the two arcs they stand on make up the whole circle and each angle is half its arc.
Another way: picture
One arc at the bottom of a circle with three different vertices on the top of it, each joined to the two ends of the arc. All three angles at the top are the same size, and the angle at the centre on the same arc is twice any of them.
Another way: story
Every one of these follows from one fact: two radii make an isosceles triangle. Draw the radii to the ends of the arc, mark the equal base angles, and the halving falls out of the angle sum.
Halving the wrong angle. The inscribed angle is half the arc, which is the same as half the angle at the centre. Two inscribed angles on the same arc are equal to each other, not half of each other.
Using the wrong pair in a cyclic quadrilateral. It is the opposite angles that add to $180^\circ$. Adjacent angles of a cyclic quadrilateral have no fixed sum at all.
Measuring the tangent angle from the wrong line. The right angle is between the tangent and the radius drawn to the point of contact — not to any other radius, and not to a chord. Draw that one radius in before you claim the $90^\circ$.
Join the centre to the two ends of the arc; the angle at the centre is the arc, $100^\circ$.
That is what measuring an arc in degrees means.
The inscribed angle on the same arc is half of it: $50^\circ$.
The angle of $110^\circ$ stands on the arc opposite it, so that arc is $220^\circ$.
Double the inscribed angle.
The rest of the circle is $360 - 220 = 140^\circ$, and the opposite angle stands on it.
Half of $140$ is $70^\circ$, and indeed $110 + 70 = 180$.
Which is the theorem, derived rather than remembered.
The angle at the centre is the whole arc.
The inscribed angle is half of it: $70^\circ$.
Put each description on the part of the circle it names.
This task has no paper form; do it on a device.
An arc of a circle measures $150^\circ$. What does an inscribed angle standing on that arc measure?
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All four corners of a quadrilateral lie on one circle, and one of its angles measures $104^\circ$. What does the opposite angle measure?
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A tangent of length $14$ cm is drawn to a circle from a point outside it, and she needs the length of the second tangent drawn from the same outside point. Which is it?
You can find the circumference $2\pi r$ and the area $\pi r^2$ of a whole circle, and you can take a fraction of a quantity. An arc and a sector need nothing more than those two things put together, because a part of a circle is a fraction of the whole one.
Arc length: the distance along the curved edge, measured in the same units as the radius.
Sector: the pie slice between two radii and the arc between them; its area is measured in square units.
Segment of a circle: the region between a chord and its arc — not the same thing as a sector.
Subtend: to make, at the centre, the angle the question is about.
Radian: the angle whose arc is exactly one radius long. A whole turn is $2\pi$ radians, so $180^\circ = \pi$ radians.
Exact form: an answer left as $6\pi$ rather than rounded to $18.85$.
An arc and a sector are the same fraction of a circle as their angle is of $360^\circ$. Arc length is that fraction of the circumference: on a circle of radius $6$, an arc of $60^\circ$ is $\frac{60}{360} \times 2\pi \times 6 = 2\pi$. Sector area is that fraction of the area: a $90^\circ$ sector of the same circle is $\frac{90}{360} \times \pi \times 36 = 9\pi$. A radian measures the same angle a different way: it is the arc length divided by the radius, so an arc of $10$ on a circle of radius $4$ subtends $2.5$ radians. A whole turn has circumference $2\pi r$, which is $2\pi$ radiuses, so a whole turn is $2\pi$ radians and $180^\circ = \pi$ radians. In radians the formulas lose their fractions altogether: arc $= r\theta$ and sector area $= \frac{1}{2}r^2\theta$.
Another way: picture
A circle cut into six equal $60^\circ$ slices. One slice is shaded: its curved edge is one sixth of the way round, and its area is one sixth of the disc.
Another way: steps
Write the fraction $\frac{\text{angle}}{360}$. Decide whether the question wants a length or an area. Multiply the fraction by $2\pi r$ for a length or by $\pi r^2$ for an area. Leave the $\pi$ in unless you are asked for a decimal.
Using the area formula for an arc. An arc is a length, so it comes from the circumference $2\pi r$; a sector is an area, so it comes from $\pi r^2$. If your answer to a length question has an $r^2$ in it, you took the wrong whole.
Forgetting the fraction. $\frac{90}{360}$ is $\frac{1}{4}$, and a quarter sector of a circle of radius $6$ has area $9\pi$, not $36\pi$. Write the fraction down before you multiply anything.
Treating a radian as a unit you convert into by dividing by something. A radian is a ratio: arc divided by radius, so an arc of $10$ on a radius of $4$ is $2.5$ radians and there is no $\pi$ anywhere in that calculation. The $\pi$ appears only when you compare with a whole turn.
The fraction of the circle is $\frac{60}{360} = \frac{1}{6}$.
Angle over a whole turn.
The circumference is $2\pi \times 6 = 12\pi$.
A length question, so start from the circumference.
$\frac{1}{6} \times 12\pi = 2\pi$.
A radian is one radius laid along the arc, so count how many radiuses fit.
$10 \div 4 = 2.5$ radians.
No $\pi$ appears, because no comparison with a whole turn was needed.
The fraction is $\frac{90}{360} = \frac{1}{4}$, and the area is $\pi \times 36 = 36\pi$.
$\frac{1}{4} \times 36\pi = 9\pi$.
A circle has radius $6$. An arc of it subtends $120^\circ$ at the centre. Its length is $k\pi$: what is $k$?
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A circle has radius $6$. A sector of it has angle $240^\circ$. Its area is $k\pi$: what is $k$?
Answer:
An arc of length $10$ lies on a circle of radius $2$. What angle does it subtend at the centre, in radians?
Answer:
A circle has radius $12$. An arc of it subtends $60^\circ$ at the centre. Its length is $k\pi$: what is $k$?
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A windscreen wiper turns through $60^\circ$ about its spindle. Its arm is $24$ cm long from the spindle to the far end of the blade, but the rubber blade itself only starts $18$ cm out. The area of glass it clears is $k\pi$ square centimetres: what is $k$?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
All four corners of a quadrilateral lie on one circle, and one of its angles measures $110^\circ$. What does the opposite angle measure?
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A circle has radius $6$. A sector of it has angle $60^\circ$. Its area is $k\pi$: what is $k$?
Answer:
You can name the parts of a circle, apply the inscribed angle, tangent and cyclic quadrilateral theorems, and compute arcs, sectors and radians. Explain why an angle in a semicircle must be a right angle, and why an arc divided by its radius is an angle at all.
8. Your turn: an arc of $140^\circ$, step 2
19. Your turn: a $90^\circ$ sector of a circle of radius $6$, step 2