Twój koszyk jest pusty
Constructing axially symmetric polygons
Example 1
Construct an isosceles triangle with a \latex{5\,cm} base and \latex{30°} base angles.
Solution
\latex{30\degree}
\latex{a= 5 {\,cm}}
Information:
Steps of construction:
Sketch:
- Construct the base of the triangle, a line segment measuring \latex{ 5\, cm\; (a = 5\, cm) }.
- Copy the \latex{30°} angle on both endpoints of side \latex{ a }.
- Extend the arms of the resulting angles. Their intersection is \latex{ A }, the third vertex of the triangle.
\latex{A}
\latex{B}
\latex{C}
\latex{a}
\latex{30\degree}
\latex{30\degree}
\latex{b}
\latex{b}
\latex{\text{\textcolor{c20100}{Construction:}}}
Discussion:
Based on the given information, the exercise has exactly one solution.
Example 2
Construct an isosceles triangle with \latex{5\,cm} long sides and \latex{45°} base angles.
Solution
\latex{45\degree}
\latex{b= 5{\,cm}}
Information:
Steps of construction:
Sketch:
- Draw a ray with starting point \latex{ B }.
- Copy the \latex{45°} angle to vertex \latex{ B } on the ray.
- Mark a \latex{ b = 5\, cm } long line segment on the arm of the angle to get vertex \latex{ A } of the triangle.
- Draw an arc with a \latex{ 5\, cm } radius around vertex \latex{ A } to mark vertex \latex{ C } of the triangle.
\latex{A}
\latex{B}
\latex{C}
\latex{a}
\latex{45\degree}
\latex{b}
\latex{b}
\latex{\text{\textcolor{c20100}{Construction:}}}
Is it true?
The resulting triangle is a right-angled triangle.
The resulting triangle is a right-angled triangle.
Discussion:
This exercise has exactly one solution.
Example 3
Construct a rhombus with \latex{5{\,cm}} and \latex{6{\,cm}} long diagonals.
Solution
During the construction, remember that the diagonals of a rhombus are the perpendicular bisectors of each other.
\latex{e= 5 {\,cm}}
Information:
\latex{f= 6 {\,cm}}
Steps of construction:
Sketch:
- Construct diagonal \latex{ f } to get vertices \latex{ A } and \latex{ C } of the rhombus.
- Construct the perpendicular bisector of line segment \latex{ AC } to get point \latex{ P }, the midpoint of line segment \latex{ AC }.
- Construct half of diagonal \latex{ e } and measure its length on the perpendicular bisector in both directions from point \latex{ P }. Now, you have vertices \latex{ B } and \latex{ D }.
- Connect points \latex{A, B, C} and \latex{D}.
\latex{A}
\latex{B}
\latex{C}
\latex{e}
\latex{ f }
\latex{D}
\latex{P}
\latex{\text{\textcolor{c20100}{Construction:}}}
Discussion:
The exercise has exactly one solution.
Example 4
Construct a kite (deltoid) with \latex{ 3 } and \latex{ 4\, cm } long sides. Its shorter diagonal measures \latex{ 4\, cm } in length.
Solution
Information:
\latex{a= 3{\,cm}}
\latex{b= 4{\,cm}}
\latex{e= 4{\,cm}}
Steps of construction:
Sketch:
- Construct diagonal \latex{ e }.
- Draw arcs with a radius of \latex{ 3\, cm } from the endpoints of diagonal \latex{ e } (points \latex{ B } and \latex{ D }). The intersection of the arcs is \latex{ A }.
- Draw arcs with a radius of \latex{ 4\, cm } from the endpoints of diagonal \latex{ e } (points \latex{ B } and \latex{ D }). The intersection of the arcs is \latex{ C }.
- Connect intersections \latex{ A } and \latex{ C } with vertices \latex{ B } and \latex{ D }.
\latex{C_1}
\latex{C_1}
\latex{B}
\latex{B}
\latex{D}
\latex{D}
\latex{C_2}
\latex{C_2}
\latex{A_2}
\latex{A_2}
\latex{a}
\latex{a}
\latex{a}
\latex{a}
\latex{e}
\latex{e}
\latex{A_1}
\latex{A_1}
\latex{b}
\latex{b}
\latex{b}
\latex{b}
\latex{\text{\textcolor{c20100}{Construction:}}}
Discussion:
The arcs drawn from points \latex{ B } and \latex{ D } have two intersections each (\latex{A_1} and \latex{A_2}, and \latex{C_1} and \latex{C_2}); therefore, using the information given, four kites can be constructed. Kites \latex{A_1BC_2D} and \latex{A_2DC_1B} are convex, while kites \latex{A_2BC_2D} and \latex{A_1DC_1B} are concave. These pairs are congruent. As a result, this exercise has exactly two solutions.
\latex{A_1}
\latex{A_1}
\latex{A_2}
\latex{A_2}
\latex{C_1}
\latex{C_1}
\latex{C_2}
\latex{C_2}
\latex{B}
\latex{B}
\latex{B}
\latex{B}
\latex{D}
\latex{D}
\latex{D}
\latex{D}
Example 5
Construct an isosceles trapezium with an \latex{ 8\, cm } long base and \latex{ 4\, cm } long sides. The angles formed by the base and the sides are \latex{60°}.
Solution
Information:
\latex{a= 8{\,cm}}
\latex{b= 4{\,cm}}
\latex{60\degree}
Steps of construction:
Sketch:
- Construct base \latex{ a }.
- Copy the \latex{60°} angle to endpoints \latex{ A } and \latex{ B }.
- Measure \latex{ 4\, cm } on the arms of the angles to get vertices \latex{ D } and \latex{ C }.
- Connect vertices \latex{D} and \latex{C}.
\latex{A}
\latex{b}
\latex{B}
\latex{C}
\latex{D}
\latex{b}
\latex{a}
\latex{c}
\latex{60\degree}
\latex{60\degree}
\latex{\text{\textcolor{c20100}{Construction:}}}
Discussion:
Using the information given, the exercise has exactly one solution.
Is it true?
The resulting trapezium can be divided into three equilateral triangles.
The resulting trapezium can be divided into three equilateral triangles.
\latex{4}
\latex{4}
\latex{4}
\latex{4}
\latex{4}
\latex{4}
\latex{4}
Exercises
{{exercise_number}}. Construct an isosceles triangle using the following information and the image.
- \latex{b = 5.2{\,cm}} b) \latex{a=6{\,cm}}
\latex{\alpha=75\degree} \latex{\alpha=120\degree}
\latex{A}
\latex{B}
\latex{C}
\latex{\beta}
\latex{\beta}
\latex{\alpha}
\latex{b}
\latex{b}
\latex{a}
{{exercise_number}}. Construct a kite with \latex{ 4\, cm } long sides and an angle of \latex{45°}. What can you say about this kite?
{{exercise_number}}. Construct a kite using the following information and the image.
- \latex{b = 4.5 {\,cm}} b) \latex{a = 3.8 {\, cm}}
\latex{f = 6 {\,cm}} \latex{\beta = 75 \degree}
\latex{\alpha = 45 \degree} \latex{\gamma = 135 \degree}
\latex{\alpha = 45 \degree} \latex{\gamma = 135 \degree}
- \latex{a = 3.4{\,cm}} d) \latex{e = 4.2 {\,cm}}
\latex{b = 5{\,cm}} \latex{f = 7 {\,cm}}
\latex{\beta = 150 \degree} \latex{e} intersects \latex{f} at one-fourth of \latex{ f }'s length
\latex{\beta = 150 \degree} \latex{e} intersects \latex{f} at one-fourth of \latex{ f }'s length
\latex{B}
\latex{D}
\latex{A}
\latex{C}
\latex{a}
\latex{a}
\latex{b}
\latex{b}
\latex{e}
f
\latex{\alpha}
\latex{\beta}
\latex{\beta}
\latex{\gamma}
{{exercise_number}}. Construct an isosceles trapezium using the following information and the image.
- \latex{a = 5 {\,cm}} b) \latex{a = 4.5 {\,cm}}
\latex{b = 3 {\,cm}} \latex{b = 3{\,cm}}
\latex{e = 4 {\,cm}} \latex{\alpha = 90 \degree}
\latex{e = 4 {\,cm}} \latex{\alpha = 90 \degree}
- \latex{b = 3.8 {\,cm}} d) \latex{b = 3 {\,cm}}
\latex{e = 4.3 {\,cm}} \latex{c = 3.5 {\,cm}}
\latex{\beta = 135 \degree} \latex{\alpha = 75 \degree}
\latex{\beta = 135 \degree} \latex{\alpha = 75 \degree}
What can you say about the isosceles trapezium in case b)?
\latex{D}
\latex{C}
\latex{B}
\latex{A}
\latex{b}
\latex{b}
\latex{a}
\latex{e}
\latex{e}
\latex{\beta}
\latex{\beta}
\latex{\alpha}
\latex{\alpha}
\latex{c}
{{exercise_number}}. Construct an isosceles trapezium with a \latex{ 6.5\, cm } long base, \latex{ 4.2\, cm } long sides and \latex{75°} base angles. Construct the perpendicular bisector of each side. What do you notice?
{{exercise_number}}. Construct the mirror image of an equilateral triangle \latex{ ABC } across
- one of its sides;
- two of its sides;;
- all three of its sides.
What type of polygons are formed by the original triangles and their mirror images in each case?
Quiz
Paul, Matt and Oliver live on the perimeter of an isosceles triangle-shaped field. The school is at the midpoint of the base, while the football field is at the vertex opposite the base. Using the image, mark where Matt and Oliver live if their houses are at the endpoints of the isosceles triangle's base.
football field
Paul’s house
school