Geometry, Plane Geometry, Symmetry

Symmetry in geometry refers to a shape or object remaining unchanged under certain transformations like reflection (line symmetry) or rotation (rotational symmetry). Questions involve identifying types of symmetry, lines of symmetry, and centers/orders of rotation for various figures.

• Question

  Divide the given shape into four congruent parts:

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  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries)

• Question

  Dissect the given shape into four congruent parts:

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  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries)

• Question

  Dissect the given shape into four congruent parts:

  

  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries)

• Question

  In the figure, a semicircle is shown with three circles inscribed within it: the large circle, tangent to the diameter of the semicircle at its center, and two smaller circles on the sides, symmetrically arranged. Which area is larger: the shaded area or the unshaded area?

  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Circles Geometry -> Plane Geometry -> Symmetry

• Quarter Yin-Yang

  The shape in the drawing is constructed as follows: Divide the diameter of a semicircle into two equal parts, and construct two additional semicircles on them as diameters: one inwards, and one outwards.

  

  Divide the shape into two congruent parts.

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Circles Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems

• Question

  It is known that all the angles of the given shape are right angles. Cut the shape into two polygons of equal area. You are only allowed to use an unmarked ruler.

  

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  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems

• Gray Area

  In the figure is a rectangle composed of 5 squares of size `2 times 2 ` and a line that intersects it diagonally.
  Find the area of the shaded region.

  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Symmetry
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 3-4 Question 3

• Question

  The numbers `1,2,3,4,5` are written at the vertices of a regular pentagon, with each number at exactly one vertex. A trio of vertices is called successful if it forms an isosceles triangle, where the number at its apex is greater than the numbers at the other two vertices, or where the number at its apex is smaller than the numbers at the other two vertices.

  Find the maximum number of successful trios that can exist.

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Geometry -> Plane Geometry -> Symmetry Minimum and Maximum Problems / Optimization Problems

• Dissect into four parts

  Geometric shapes are called congruent if they coincide when superimposed. Cut the following shape into four congruent parts:

  

  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries)
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4 Question 3

• Cut a Boat in Half

  Geometric shapes are called congruent if they coincide when placed on top of each other. Cut the shape into two congruent parts

  

  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 Question 6