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Using the ruler and compass
Euclid laid down the foundations of geometry around \latex{ 300 } BC. He grouped concepts and propositions together and set out the requirements for geometric constructions.
The required instruments of Euclidean constructions
In Euclidean constructions, only a compass and a straight edge may be used. The construction is limited to the following steps:
Draw a line between two points by joining them with a straight edge.
\latex{ A }
\latex{ B }
\latex{ e }
Set the compass to the distance of the two points.
\latex{ A }
\latex{ B }
Draw a circle starting from any given point with the radius previously set.
\latex{ O }
You can determine the intersection point of two lines.
\latex{ M }
\latex{ e }
\latex{ f }
You can determine the intersection points of a circle and a line.
\latex{ O }
\latex{ e }
\latex{ A }
\latex{ B }
\latex{ k }
Intersection points \latex{ A } and \latex{ B }.
You can determine the intersection points of two circles.
\latex{ O_1 }
\latex{ A }
\latex{ B }
\latex{ k_1 }
Intersection points \latex{ A } and \latex{ B }.
\latex{ k_2 }
\latex{ O_2 }
Some basic construction steps
Constructing a \latex{ 2 \,cm } line segment:
\latex{ A }
Draw a line and mark a point
\latex{ A } on it.
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
Take up \latex{ 2 \,cm } with the compass.
\latex{ A }
Cross the line with a circle of
2 cm radius from centre A.
\latex{ A }
\latex{ B }
The point of intersection, \latex{ B } is is the other endpoint of the segment.
Drawing a circle with a \latex{ 1 \,cm } radius around any given point:
\latex{ P }
Mark an arbitrary point \latex{ P }.
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
Set the compass to \latex{ 1 \,cm }.
\latex{ P }
Draw a circle with the compass centred at \latex{ P }.
Steps of solving construction tasks
I. Making a sketch
After reading the exercise carefully, write down all the data available, then draw a freehand sketch of the design. Highlight the known elements on the sketch and examine how they relate to what you are trying to find.
After reading the exercise carefully, write down all the data available, then draw a freehand sketch of the design. Highlight the known elements on the sketch and examine how they relate to what you are trying to find.
II. Planning the design process
Identify the relationships between the information you have and the properties of the shape you want to construct. Plan the steps of the process. Number these steps on your freehand sketch and write a description of the construction process.
Identify the relationships between the information you have and the properties of the shape you want to construct. Plan the steps of the process. Number these steps on your freehand sketch and write a description of the construction process.
III. Performing the construction
Carry out the construction plan. Use only a compass and a straight edge.
IV. Analysis of the solution (discussion)
Examine whether the task can be solved. If so, consider how many solutions there are. If there is no solution, explain the reasons.
Examine whether the task can be solved. If so, consider how many solutions there are. If there is no solution, explain the reasons.
Example 1
Construct an isosceles triangle in which
- the base is \latex{ 5 \,cm } long, and the two sides are \latex{ 3 \,cm } long;
- the base is \latex{ 5 \,cm } long, and the two sides are \latex{ 2.5 \,cm } long;
- the base is \latex{ 5 \,cm } long, and the two sides are \latex{ 2 \,cm } long.
Solution (a)
Data:
\latex{a = 5 \;cm}
\latex{b = 3 \; cm}
Sketch:
Steps of construction:
- Draw a line, then take a \latex{ 5 \,cm } line segment with endpoints \latex{ B } and \latex{ C }. These will be the vertices of the triangle.
- The third vertex of the triangle, \latex{ A }, is 3 cm away from vertex \latex{ B }. Draw a circle with a radius of 3 cm with centre \latex{ B }.
- Vertex \latex{ C } is also \latex{ 3 \,cm } away from vertex \latex{ A }. Repeat the previous step but this time around vertex \latex{ C }.
- Connect the intersection of the two circles to get vertex \latex{ A }. Connect points \latex{ A, B } and \latex{ C } to form the triangle.
\latex{\text{\textcolor{c20100}{Construction:}}}
\latex{ A_1 }
\latex{ A_2 }
\latex{ B }
\latex{ C }
\latex{ b }
\latex{ b }
\latex{ a }
Discussion:
The two circles intersect at two points, \latex{ A_1 } and \latex{ A_2 }, so there are two congruent triangles: \latex{ A_1BC } and \latex{ A_2BC }. By convention, we do not consider congruent solutions to be different solutions.
Solution (b), (c)
You have to construct isosceles triangles, so the steps of construction are the same as in case a).
b)
\latex{a = 5 \;cm}
\latex{b = 2,5 \;cm}
\latex{\text{\textcolor{c20100}{Construction:}}}
\latex{ B }
\latex{ A }
\latex{ C }
c)
\latex{a = 5 \; cm}
\latex{b = 2 \; cm}
\latex{\text{\textcolor{c20100}{Construction:}}}
\latex{ B }
\latex{ C }
Discussion:
In case b), the two circles are tangent. As their common point \latex{ A } is on segment \latex{ BC }, no triangle is formed. The task has no solution.
In case c), the two circles have no common points. Therefore, the task has no solution.
From these examples, you can conclude that three sides can form a triangle only if the sum of the lengths of any two sides is greater than the length of the third side.
Exercises
{{exercise_number}}. Mark two points in your notebook: \latex{ A } and \latex{ B }. Set the compass to the distance between \latex{ A } and \latex{ B }, then draw a circle centred at \latex{ A } and another circle centred at \latex{ B }. How many intersections do the two circles have?
{{exercise_number}}. Draw two points, \latex{ A }and \latex{ O }, in your notebook. Set the compass to the distance of the two points, then draw circles with that radius in accordance with the picture. (→)
- How long are the sides of the triangle \latex{ OAB }?
- Look for isosceles triangles, rhombuses, deltoids, isosceles trapezoids and rectangles in the picture so that their vertices are the points marked by the letters (e.g. the quadrilateral \latex{ EFAD } is an isosceles trapezoid).
\latex{ C }
\latex{ B }
\latex{ D }
\latex{ O }
\latex{ A }
\latex{ F }
\latex{ E }
{{exercise_number}}. The drawing shows a stained glass window. Use the steps of construction to design a similar one. (→)
{{exercise_number}}. Construct an equilateral triangle with sides measuring \latex{ 5 \,cm }.
{{exercise_number}}. Construct an axially symmetrical triangle with side lengths of \latex{ 2 \,cm } and \latex{ 4 \,cm }.
{{exercise_number}}. The perimeter of an axially symmetrical triangle is \latex{ 15 \,cm }, and its leg is twice as long as its base. Calculate the lengths of the sides, then use a compass to mark these lengths and construct the triangle.
{{exercise_number}}. Construct an isosceles triangle with legs that are the same length as segment \latex{ b }, and a perimeter equal to segment \latex{ k }.
\latex{ b }
\latex{ k }
{{exercise_number}}. The perimeter of an isosceles triangle is \latex{ 14 \,cm }, and the lengths of its sides (in centimetres) are whole numbers. How long can the base of the triangle be? How many such triangles can be formed? Construct the triangles.
{{exercise_number}}. Construct triangles using the segments given:
\latex{ a }
\latex{ b }
\latex{ c }
How many triangles can you construct if the sides of the triangles can be equal?
Quiz
Continue the sequence with three additional terms.
