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The circle
A circle is a set of points in the plane found at the same distance from a given point.
The given point is the centre of the circle \latex{ (O) }, while the distance is the radius \latex{ (r) }.
Terminology related to circles
Radius: a line segment connecting the centre and any point of the circle.
Diameter: a line segment passing through the centre and connecting two points of the circle.
Symbol: \latex{ d }.
Diameter: a line segment passing through the centre and connecting two points of the circle.
Symbol: \latex{ d }.
The length of the diameter is two times the length of the radius \latex{(d=2\times r)}.
Arc: a segment of the circle.
Disc: a plane figure bounded by a circle.
Sector: the area between two radii and their connecting arc.
Disc: a plane figure bounded by a circle.
Sector: the area between two radii and their connecting arc.
Circle and radius
circle
\latex{ O }
radius \latex{ (r) }
diameter
\latex{ O }
\latex{ O }
arc
disc
\latex{ O }
\latex{ O }
sector
Example 1
Colour the parts of the plane that are
a) exactly \latex{ 1\,cm } away from point \latex{ O };
b) at most \latex{ 1\,cm } away from point \latex{ O };
c) at least \latex{ 1\,cm } away from point \latex{ O }.
a) exactly \latex{ 1\,cm } away from point \latex{ O };
b) at most \latex{ 1\,cm } away from point \latex{ O };
c) at least \latex{ 1\,cm } away from point \latex{ O }.
Solution
a)
b)
c)
\latex{ O }
\latex{ O }
\latex{ O }
circle
disc
plane
- The set of points \latex{ 1\,cm } away from point \latex{ O } is a circle with a radius of \latex{ 1\,cm }.
- The set of points found at a maximum distance of \latex{ 1\,cm } from point \latex{ O } is a disc with a radius of \latex{ 1\,cm } (including the circle itself).
- The set of points found at a minimum distance of \latex{ 1\,cm } from point \latex{ O } is a plane with a \latex{ 1 }-\latex{ cm }-radius disc cut out of it.
The word circle may refer to the disc or the line bordering it.
Constructing a circle
Construct a circle with a centre \latex{ O } and radius \latex{ r }.
- You have point \latex{ O } and line segment \latex{ r }.
- Open your compass to a distance equal to the length of line segment \latex{ r }.
- Place the point of the compass at point \latex{ O }.
- Draw a circular line.
\latex{ r }
\latex{ O }
\latex{ O }
\latex{ r }
\latex{ O }
\latex{ O }
\latex{ O }
\latex{ r }
\latex{ r }
\latex{ r }
Example 2
Grandpa turned the sprinkler on in the middle of the garden. It sprinkles water to a distance of \latex{ 3\,m }. Where can Grandpa sit to read the newspaper if he does not want to get wet? (The garden is square-shaped, the sides are \latex{ 10\,m } long and the sprinkler is in the middle.)
Solution
Draw the floor plan of the garden with the sprinkler. (In your notebook, \latex{ 1\,cm } should correspond to \latex{ 1\,m }.) Since water is sprinkled to a distance of \latex{ 3\,m }, construct a circle with a centre \latex{ S } and a radius of \latex{ 3\,cm } (point \latex{ S } is the centre of the square).
If Grandpa sits inside the circle, he will get wet. Therefore, he has to sit outside this circle.
\latex{ S }
\latex{ 3\,m }
\latex{ 10\,m }
Example 3
Find a point \latex{ M } in the plane which is \latex{ 20\,mm } away from point \latex{ K } and \latex{ 10\,mm } away from point \latex{ L } if the length of the line segment \latex{ KL } is
a) \latex{ KL = 25\,mm }; b) \latex{ KL = 30\,mm }; c) \latex{ KL = 40\,mm }.
Solution
Since the distance between point \latex{ M } and point \latex{ K } is \latex{ 20\,mm }, point \latex{ M } lies on a circle with a radius of \latex{ 20\,mm } and a centre \latex{ K }. Since the distance between point \latex{ M } and point \latex{ L } is \latex{ 10\,mm }, point \latex{ M } lies on a circle with a radius of \latex{ 10\,mm } and a centre \latex{ L }. The intersections of the two circles are the points you are looking for.
\latex{ L }
\latex{ K }
\latex{ M_{1} }
\latex{ M_{2} }
Steps of the geometric construction:
- Draw segment \latex{ KL }.
- Construct the circle with centre \latex{ K } and a radius of \latex{ 20\,mm }.
- Construct the circle with centre \latex{ L } and a radius of \latex{ 10\,mm }.
- Mark the points of intersection.
\latex{\textcolor{#d3d3d3}{K}}
\latex{\textcolor{#d3d3d3}{L}}
\latex{ K }
\latex{ L }
\latex{ K }
\latex{ L }
\latex{ K }
\latex{ L }
In case a), there are two solutions. Since the two circles intersect, the circles have two common points. If you connect points \latex{ K }, \latex{ L }, and \latex{ M_{1} }, you get a triangle.
In case b), there is one solution. Since the two circles are tangent, they have one common point. Points \latex{ K }, \latex{ L }, and \latex{ M } are on the same line.
In case c), there is no solution because the two circles do not have a common point.
a)
b)
c)
\latex{ K }
\latex{ K }
\latex{ K }
\latex{ L }
\latex{ L }
\latex{ L }
\latex{ M_{1} }
\latex{ M_{2} }
\latex{ M }
Exercises
{{exercise_number}}. Draw some of the circle-shaped traffic signs.
{{exercise_number}}. Copy the following figures to graph paper and colour them.
{{exercise_number}}. How many common points can the following geometric shapes have?
a) a line and a circle
b) a line and a disc
c) the sides of a triangle and a circle
{{exercise_number}}. How many common points can two circles with
a) different radii;
b) the same radius have?
{{exercise_number}}. According to the image, the common point of circles, \latex{ k }, \latex{ m } and \latex{ n } is point \latex{ C }. The centre of circle \latex{ k } lies on circle \latex{ m }, while the centre of circle \latex{ m } is found on circle \latex{ n }. What is the diameter of circle \latex{ k } if the radius of circle \latex{ n } is \latex{ 3\,cm? } (→)
\latex{ k }
\latex{ m }
\latex{ n }
\latex{ C }
{{exercise_number}}. Construct points \latex{ 3\,cm } away from one of the ends of line \latex{ EF = 5\,cm } and \latex{ 4\,cm } away from the other end. What shape do you get if you connect the constructed points with points \latex{ E } and \latex{ F? }
{{exercise_number}}. \latex{ 1 }-\latex{ year }-old Pete can throw his toys \latex{ 1\,m } away from his playpen. Draw the floor plan of the room and mark where the toys can land. (The sides of the playpen are \latex{ 1\,m } × \latex{ 1.5\,m }, while the sides of the square room are \latex{ 4\,m } long.)
{{exercise_number}}. How many parts can a plane be divided into by
a) \latex{ 1 };
b) \latex{ 2 };
c) \latex{ 3 } circles?
{{exercise_number}}. Grab a sheet of paper, fold it in half, and punch a hole outside the fold using a compass. Unfold the paper, and mark the two points made by the holes \latex{ A } and \latex{ B }. Construct line segment \latex{ AB }.
a) Mark three arbitrary points \latex{ (E, F, G) } along the fold and measure their distance from points \latex{ A } and \latex{ B }. What do you notice?
b) What is the relative position of the fold and line segment \latex{ AB }?
Quiz
How many common points can three discs have?
