Euclidean Geometry
Collection of Euclidean Geometry worksheets.
Table des matières
- Proofs
- The (mathematical) logic behind the scenes - Lesson+Exploration+Practice
- Counterexamples - Lesson+Exploration
- Napoleon's Theorem - Defining Hypotheses and Thesis - Lesson+Exploration
- Triangles, Squares and Unexpected Areas
- Measuring
- Subtracting Hours and Minutes - Practice
- Angles, Segments and Lines
- Relative Positions of Two Lines in the Plane - Lesson+Exploration+Practice
- Angles Between Two Lines and a Transversal - Lesson+Exploration+Practice
- Parallel lines, Transversals and Segments
- Thales' Intercept Theorem
- Construcing a 30° Angle + Properties of Rhombi
- Construcing a 60° Angle + Properties of 30°-60°-90° Triangles
- Constructing the Angle Bisector + Activities
- Constructing the Copy of an Angle + Angles Activities
- Constructing the Midpoint and the Perpendicular Bisector of a Segment
- Constructing the Perpendicular Line Through a Point on the Line
- Constructing the Perpendicular Line Through a Point Not on the Line
- Triangles
- Classifying Triangles
- Special Points of Triangles - Lesson+Exploration+Practice
- Special Right Triangles 30°-60°-90° and 45°-45°-90°
- Constructing an Isosceles Triangle with Given Side + Activities
- Constructing Equilateral Triangles + Activities
- Constructing an Inscribed Equilateral Triangle + Activities
- Constructing a triangle, given a side and two angles
- Pythagorean Theorem - a "proof without words"
- Dissections - Haberdasher's puzzle
- Golden Triangle and Spiral
- Polygons
- Convex, Concave and Self-Intersecting Polygons - Lesson+Exploration+Practice
- Exploring rotational symmetry of regular polygons - Lesson+Exploration+Practice
- Rotational symmetry and lines of symmetry of a regular polygon - Lesson+Exploration+Practice
- Constructing a Regular Hexagon + Practice
- Regular Star Polygons
- Circles
- The Gardener's Circle
- Construction of the Circle Passing Through 3 Points
- Parts of a Circle
- Circles - Relative Positions
- Areas
- Triangle Area = (b ∙ h)/2 Why? Visual proof (1)
- Triangle Area = (b ∙ h)/2 Why? Visual proof (2)
- Regular Polygons and Equivalent Triangles
- Pythagoras, semicircles and lunes - Lesson+Exploration
- Misc.
- Kaleidoscope
- Ruotiamo...
The (mathematical) logic behind the scenes - Lesson+Exploration+Practice
Material implication: ⇒
Let's consider the statement [math]p:x=3[/math], and the statement [math]q:x+1=4[/math].[br]We say that [math]p[/math] [i]implies [/i][math]q[/math], and we write [math]p\Longrightarrow q[/math] to mean that if [math]p[/math] is true, then also [math]q[/math] is true. [br](If [math]p[/math]is true, [math]q[/math] is [i]necessarily [/i]true). [br][br]The symbol [math]\Longrightarrow[/math] [i]connects [/i]a premise [math]\left(p\right)[/math] and a conclusion [math]\left(q\right)[/math] and is very used in proofs, because it's a symbolic way to show deductive reasoning.[br][br]The statement "[math]p[/math] implies [math]q[/math]" is also written "if [math]p[/math] then [math]q[/math]" or sometimes "[math]q[/math] if [math]p[/math]".[br][br]Does this sound complicated? No... let's see a few examples of implications.[br][list][*][i]If [/i]you score 68% or more in this problem, [i]then [/i]you will pass the exam.[/*][*]Your head will hurt [i]if[/i] you bang it against a wall.[/*][/list]
Exploring implications in geometry
Given a quadrilateral [i]Q[/i], use the applet below to find out the reciprocal implications between the following statements:[br][i]a[/i]: [i]Q[/i] has an obtuse angle.[br][i]b[/i]: [i]Q[/i] has three acute angles.[br][i]c[/i]: [i]Q[/i] has no right angles. [br][br](drag the orange points to explore different quadrilaterals)
Which are the implications between statements [i]a[/i], [i]b[/i] and [i]c[/i]?
Implication is confused by fake guys
Consider this example: [math]10=11\Longrightarrow10\cdot0=11\cdot0[/math][br]We started with a false premise and implied a true conclusion.[br][br]Now consider this: [math]10=11\Longrightarrow10-1=11-1[/math][br]We started - again - with a false premise, and implied a wrong conclusion.[br][br]Implication doesn't like false premises. If we start with a false premise, the conclusion obtained by implication can be anything.
Showing why things go wrong
In the example above, we had the following three statements about a quadrilateral [i]Q[/i]:[br][i]a[/i]: [i]Q[/i] has an obtuse angle.[br][i]b[/i]: [i]Q[/i] has three acute angles.[br][i]c[/i]: [i]Q[/i] has no right angles. [br][br]We can say that:[br][list][*][i]a[/i] doesn't imply [i]b [/i]because a rhombus (that is not a square) has an obtuse angle, but not 3 acute ones.[/*][*][i]a[/i] doesn't imply [i]c[/i] because a right trapezoid (that is not a rectangle) contains an obtuse angle, and two right angles.[/*][*][i]c[/i] doesn't imply [i]b[/i] because a rhombus (that is not a square) has no right angles, but doesn't have three acute angles.[/*][/list][br]We explained that implication doesn't hold using a "tool" that in mathematics is named [i]counterexample[/i].[br]Click [url=https://www.geogebra.org/m/aexepwzu]here [/url]to discover more about counterexamples.
Subtracting Hours and Minutes - Practice
Practice subtracting hours and minutes, and become a time master!
The hint and solution displayed in the app show a quick way to subtract hours and minutes.[br]How would you calculate the difference in a different way?[br]Explain your method, describing it step by step.
Relative Positions of Two Lines in the Plane - Lesson+Exploration+Practice
In the app below, select the relative position that you want to explore, and change the displayed straight lines by dragging the points that define them.
Answer the following questions
There is only one line passing through three distinct points in a plane.
Given two points in a plane, there are infinitely many distinct lines passing through them.
Any two lines intersect at a point.
Classifying Triangles
Explore...
Discover how to classify triangles by sides or by angles, then answer the following questions.[br][br][i]Drag the vertices to explore different triangles.[/i]
True or false?
Determine whether the following statement is true or false. [br]If it is true, draw a triangle that satisfies the given condition; if it is false, explain the reason for your answer, possibly by providing a counterexample.[br][br]"[i]A scalene triangle cannot be obtuse[/i]."
Determine whether the following statement is true or false. [br]If it is true, draw a triangle that satisfies the given condition; if it is false, explain the reason for your answer, possibly by providing a counterexample.[br][br]"[i]An equilateral triangle can be obtuse[/i]."
Determine whether the following statement is true or false. [br]If it is true, draw a triangle that satisfies the given condition; if it is false, explain the reason for your answer, possibly by providing a counterexample.[br][br]"[i]A scalene triangle can be a right triangle[/i]."
Classification and Sets
Consider the following sets:[br][i]T[/i] = {[i]x[/i] | [i]x[/i] is a triangle} [br][i]E[/i] = {[i]x[/i] | [i]x[/i] is an equilateral triangle} [br][i]I[/i] = {[i]x[/i] | [i]x[/i] is an isosceles triangle} [i][br]S[/i] = {[i]x[/i] | [i]x[/i] is a scalene triangle} [br]Represent the classification of triangles by sides, using Venn diagrams.
Consider the following sets:[br][i]T[/i] = {[i]x[/i] | [i]x[/i] is a triangle} [br][i]R[/i] = {[i]x[/i] | [i]x[/i] is a right triangle} [br][i]A[/i] = {[i]x[/i] | [i]x[/i] is an acute triangle} [i][br]O[/i] = {[i]x[/i] | [i]x[/i] is an obtuse triangle} [br]Represent the classification of triangles by angles, using Venn diagrams.
Convex, Concave and Self-Intersecting Polygons - Lesson+Exploration+Practice
Simple and self-intersecting polygons
In the app below you can see a [i]polygon: [/i]polygon is a word derived from Greek, and means "many angles".[br][br]A polygon can be [i]simple[/i], when its boundary does not cross itself, otherwise it's [i]self-intersecting[/i].[br][br]Drag the vertices of the polygon (green points) and create [i]simple[/i] and [i]self-intersecting[/i] figures. The intersection points will be shown in orange.[br]
Convex and Concave Polygons
When polygons are [i]simple[/i], we can characterize them further, as [i]convex [/i]or [i]concave[/i].[br][br]Explore the applet below, create a few different [i]convex [/i]and [i]concave[/i] polygons, observe the measure of the angles and the position of the diagonals with respect to the portion of the plane enclosed within its sides.[br][br]Formulate your own conjecture about how to differentiate a [i]convex [/i]and a [i]concave [/i]polygon, depending on these properties of the figure.
Convex polygons and interior angles
What can you say about the measures of the interior angles of a [i]convex [/i]polygon?[br]Do they all have a common characteristic?
Concave polygons and interior angles
What can you say about the measures of the interior angles of a [i]concave[/i] polygon?[br]Do some of them have a common characteristic?
Convex polygons, concave polygons and their diagonals
Describe the [i]difference [/i]between the [i]position[/i] of the [i]diagonals[/i] of a [i]convex [/i]and of a [i]concave [/i]polygon, with respect to the portion of the plane enclosed within its sides.
The Gardener's Circle
How can a gardener draw a perfectly round contour for a flowerbed?[br][br]Two stakes and a rope are enough: one stake is fixed at the desired position of the center of the flowerbed, one end of the rope is attached to the fixed stake, and the other end is attached to the second stake.[br][br]At this point, the gardener can mark the contour of the flowerbed using the free stake, moving around the fixed one while making sure that the rope is always taut.[br][br][i]Use the app below to create your virtual flowerbed using the gardener's method. The slider allows you to change the length of the rope. [br]Drag point P or the tip of the stake in the 3D View to create the flowerbed. [br][br]The 3D view can be rotated by dragging it with the mouse or using the predefined gestures for touch screens, and the buttons allow you to restore the 3D view to the original settings and delete the traces.[/i]
What common characteristic do all the points on the flowerbed's contour have?
In geometric terms, the contour of the flowerbed is a circle. [br]How would you define the rope, using the same language?
If the gardener uses a rope that is 2 meters long, would a mushroom 3 meters away from the fixed stake be inside, outside, or on the contour of the flowerbed?[br]What about a flower 1.9 meters away from the fixed stake?[br]And a stone 2 meters away from the fixed stake?[br]After writing down your answers, explain them using the language of geometry.
Triangle Area = (b ∙ h)/2 Why? Visual proof (1)
Let's define first the objects we will be working with.[br]We say that two figures [math]F_1[/math] and [math]F_2[/math] are [i][color=#1e84cc]equidecomposable [/color][/i]if they can be dissected into a finite number of parts, respectively congruent.[br][br]This means that if we have two figures, and we are able to cut each of them such that each cut out piece is the same for both figures, then the two of them are said to be "[i]equidecomposable[/i]".[br]The figures that you can build using all the pieces of a [url=https://www.geogebra.org/m/kyzukdsr#material/hue9x3yf]Tangram game[/url] are an example of [i]equidecomposable [/i]figures.[br][br]In the app below you will discover why, given any triangle, you can calculate its area by multiplying its base by half of its height.[br][br]Use the [color=#ff7700]slider [/color]to dissect (cut) the triangle and obtain an equivalent parallelogram.[br][br]Move the vertices [i][color=#6aa84f]A[/color][/i], [i][color=#6aa84f]B[/color][/i] and [i][color=#6aa84f]C[/color][/i] of the triangle to explore different configurations.
Kaleidoscope
Drag the red dots and the central red small circle to obtain a nice kaleidoscope effect.