Twój koszyk jest pusty
Lines of symmetry in triangles
Perpendicular bisector
Point \latex{ A } and axis \latex{ l } are constructed. Find the mirror image of point \latex{ A } across axis \latex{ l } and mark it as point \latex{ B }. These two points form the line segment \latex{ AB }.
Connect points \latex{ A } and \latex{ B }. Denote the intersection of line segment \latex{ AB } and axis \latex{ l }by \latex{ F }. Point \latex{ A } and point \latex{ B } are the same distance from axis \latex{ l }; thus, \latex{ F } is also the midpoint of line segment \latex{ AB }. Axis \latex{ l } is perpendicular to line segment \latex{ AB }.
Axis \latex{ l } is the perpendicular bisector of line segment \latex{ AB }.
\latex{A}
\latex{B}
\latex{l}
\latex{A}
\latex{B}
\latex{l}
\latex{F}
The perpendicular bisector of a line segment is a line that meets the line segment at its midpoint perpendicularly.
Properties of the perpendicular bisector
Connect the endpoints of line segment \latex{AB} with points \latex{C,D} and \latex{E} on its perpendicular bisector.
Since the perpendicular bisector is also the line of symmetry of the line segment \latex{ AB }, the resulting line segments are mirror images of each other; thus, they are equal in length:
\latex{ CA = CB = c,}
\latex{ DA = DB = d,}
\latex{ DA = DB = d,}
\latex{EA = EB = e.}
If a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of that line segment.
\latex{A}
\latex{B}
\latex{c}
\latex{D}
\latex{E}
\latex{l}
\latex{C}
\latex{c}
\latex{d}
\latex{d}
\latex{e}
\latex{e}
If a point is not on the perpendicular bisector of a line segment, then it is not equidistant from the endpoints of that line segment.
For example:
\latex{ GA\lt GB, }
\latex{ HA\gt HB. }
\latex{ HA\gt HB. }
If a point is equidistant from the endpoints of a line segment, it is on the perpendicular bisector of that line segment.
\latex{G}
\latex{A}
\latex{B}
\latex{l}
\latex{H}
These points are closer
to A than B.
to A than B.
These points are closer
to B than A.
to B than A.
Isosceles triangles
Axis \latex{ l } and line segment \latex{ AB } are constructed. One of the endpoints of the line segment (point \latex{ A }) is on the axis. Find the mirror image of \latex{ AB } across the axis.
The mirror image of point \latex{ A } is itself. Denote the mirror image of point \latex{ B } by \latex{ C }.
Connect points \latex{A,B} and \latex{C}. The sides \latex{AB} and \latex{AC} of the resulting triangle \latex{ ABC } are equal in length.
\latex{A}
\latex{B}
\latex{l}
\latex{C}
A triangle with two equal sides is called an isosceles triangle.
Terminology:
- The two equal sides of an isosceles triangle are called legs (\latex{\large a}), and the third side is called the base (\latex{\large b}).
- The angle formed by the two legs is called the vertex angle (\latex{\alpha}), and the other two angles are called the base angles (\latex{\beta}).
\latex{\alpha}
\latex{\beta}
\latex{\beta}
\latex{A}
\latex{C}
leg
\latex{B}
\latex{l}
leg
base
Properties of an isosceles triangle:
- It has one line of symmetry.
- Its base angles are equal.
- Its line of symmetry is the perpendicular bisector of its base.
- Its line of symmetry divides the triangle.
The line of symmetry \latex{ l } is the perpendicular bisector of the base and the angle bisector of the vertex angle.
\latex{\alpha}
\latex{\alpha}
\latex{\beta}
\latex{\beta}
\latex{ C }
\latex{ A }
\latex{ B }
\latex{\frac{b}{2} }
\latex{\frac{b}{2} }
The angle bisector is a ray that divides a given angle into two equal angles.
\latex{\frac{\alpha}{2}}
\latex{\frac{\alpha}{2}}
\latex{\alpha}
angle bisector
\latex{ A }
Equilateral triangles
\latex{ \alpha }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ \alpha }
\latex{ \alpha }
A triangle with three equal sides is called an equilateral triangle.
An equilateral triangle has
- three lines of symmetry
- and three equal angles.
\latex{\alpha}
\latex{\alpha}
\latex{\alpha}
When three congruent equilateral triangles are placed next to each other, as shown in the image, their adjacent angles form a straight angle (\latex{ 180° }).
\latex{\alpha+\alpha+\alpha=180°.}
Thus, \latex{\alpha} \latex{ = 60° }.
\latex{\alpha}
\latex{\alpha}
\latex{\alpha}
Each angle of an equilateral triangle is \latex{ 60° }.
If each angle of a triangle is \latex{ 60° }, then it is an equilateral triangle.
\latex{ 60° }
\latex{ 60° }
\latex{ 60° }
Equilateral triangle
Triangles
Isosceles triangles
Every equilateral triangle is isosceles, but not every isosceles triangle is equilateral.
Exercises
{{exercise_number}}. You fold a piece of paper in half and cut the following figures out of the folded paper. In which cases do you get isosceles triangles?
\latex{ 1. }
\latex{ 2. }
\latex{ 3. }
\latex{ 4. }
\latex{ 5. }
\latex{ 6. }
fold
{{exercise_number}}. How many isosceles triangles can you make if they are made up of the following number of matchsticks?
- \latex{ 3 }
- \latex{ 4 }
- \latex{ 5 }
- \latex{ 6 }
- \latex{ 9 }
- \latex{ 11 }
- \latex{ 15 }
{{exercise_number}}. How can a piece of paper be cut into two parts by a single cut so that the two parts can be assembled into an isosceles triangle without overlapping?
{{exercise_number}}. How many isosceles triangles can you see in the image?
{{exercise_number}}. Four congruent isosceles triangles can form a square. What can you tell about their angles?
{{exercise_number}}. What is the perimeter of an equilateral triangle if its side is \latex{ 2,006 } \latex{ cm } long?
{{exercise_number}}. How long is the side of an equilateral triangle if its perimeter is \latex{ 2,007 } \latex{ cm ?}
{{exercise_number}}. The table contains information about isosceles triangles. Based on the given data, complete the table by writing the missing values into the blank spaces.
base (\latex{cm})
leg (\latex{cm})
perimeter (\latex{cm})
\latex{ 11 }
\latex{ 8 }
\latex{ 8 }
\latex{ 18 }
\latex{ 18 }
\latex{ 8 }
\latex{ 5 }
\latex{ 7.5 }
\latex{ 5 }
\latex{ 20 }
\latex{ 9 }
\latex{ 8 }
\latex{ 9 }
\latex{ 21 }
{{exercise_number}}. The perimeter of an isosceles triangle is \latex{ 19 } \latex{ cm }, and its base is \latex{ 1 } \latex{ cm } longer than its legs. How long are its legs?
{{exercise_number}}. The perimeter of an isosceles triangle is \latex{ 20 } \latex{ cm }, and its legs are \latex{ 1 } \latex{ cm } longer than its base. How long is its base?
{{exercise_number}}. What is the perimeter of an isosceles triangle if one of its sides is \latex{ 8 } \latex{ cm } and the other side is \latex{ 11 } \latex{ cm } long?
{{exercise_number}}. What is the perimeter of an isosceles triangle if one of its sides is \latex{ 6.4 } \latex{ cm } and the other is half of that?
{{exercise_number}}. The perimeter of a triangle is \latex{ 8.4 } \latex{ cm }, one of its sides is \latex{ 3.6 } \latex{ cm }, and its other side is one-third of that. Is the triangle isosceles?
{{exercise_number}}. An isosceles triangle was folded in half, resulting in the triangle shown in the image. What is the perimeter of the original triangle?
\latex{ 5 } \latex{ cm }
\latex{ 4 } \latex{ cm }
\latex{ 3 } \latex{ cm }
{{exercise_number}}. Determine whether the following statements are true or false.
- There is a triangle that has three lines of symmetry.
- Every triangle that has a line of symmetry is isosceles.
- There are isosceles triangles that have obtuse angles.
- If a triangle has three lines of symmetry, then each of its angles is \latex{ 60° }.
- The line of symmetry of an isosceles triangle divides the triangle into two right-angled triangles.
- If a triangle has a line of symmetry, then it also has a right angle.
{{exercise_number}}. Plot the following points in a Cartesian coordinate system:
\latex{A(-1;3)}
\latex{B(2;0)}
\latex{C(2; 6)}
\latex{D(4; 3)}
\latex{E(4; -1)}
\latex{F(6; 0)}
\latex{G(6; 4)}
- How many isosceles triangles are there, with each vertex being one of the plotted points?
- Use their coordinates to plot points that create isosceles triangles with points D and E.
{{exercise_number}}. Construct a Cartesian coordinate system.
- Construct a triangle \latex{ABC} with a red pencil. The coordinates of its vertices are the following:
\latex{A(5; 0);}
\latex{B(0; 5);}
\latex{C(0; 0);}
- Mark the mirror images of the vertices.
- Mark the points (\latex{ A_{1} } , \latex{ B_{1} } and \latex{ C_{1} } ) that are reflected across axis \latex{ x } with a blue pencil,
- and the points (\latex{ A_{2} } , \latex{ B_{2} } and \latex{ C_{2} } ) that are reflected across axis \latex{ y } with a green pencil.
Is there a vertex that has been marked with all three colours?
- What kind of quadrilateral is \latex{ ABA_{2}B_{1} } ?
{{exercise_number}}. What can you tell about a triangle if one of its angles is three times larger than the other and the third one is one-fourth of the sum of the other two angles?
Quiz
Move three of the points in the figure to get a congruent equilateral triangle but in a different position.
