Geometry

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues. Expected questions involve calculating lengths, angles, areas, and volumes of various shapes, understanding geometric theorems, and solving problems related to spatial reasoning.

Solid Geometry / Geometry in Space Trigonometry Spherical Geometry Plane Geometry Vectors

• Question

  Cut a square into three parts, and then assemble them into an acute triangle with all sides of different lengths.

  Topics:

  Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems

• Question

  Given a sheet of paper of size `10×10` cm. Can you cut out a number of circles from this sheet such that the sum of their diameters is greater than `5` meters?

  Topics:

  Arithmetic Geometry -> Area Calculation Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems

• Question

  Cut the given shape into "T" shapes as shown in the diagram:

  

  Topics:

  Geometry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry

• Question

  A plane is colored with two colors (that is, every point on the plane is colored with one of these two colors). Prove that there exist two points on the plane at a distance of `1` such that they are both the same color.

  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Combinatorial Geometry Geometry -> Plane Geometry -> Triangles Proof and Example -> Constructing an Example / Counterexample Proof and Example -> Proof by Contradiction

• Question

  Is there a quadrilateral that can be cut into `6` parts by two straight cuts? Justify your answer or provide an example.

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  Topics:

  Geometry -> Plane Geometry Proof and Example -> Constructing an Example / Counterexample 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

• Question

  In the plane, a point and `12` lines passing through it are given. Prove that there are two of these lines such that the angle between them is less than `17^@`.

  Topics:

  Combinatorics -> Pigeonhole Principle Geometry -> Plane Geometry -> Angle Calculation

• Question

  Given `12` intersecting lines in the plane. Prove that there exist two of these lines such that the angle between them is less than `17^@`.

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  Topics:

  Combinatorics -> Pigeonhole Principle Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Parallel Translation / Translation

• Question

  Two congruent triangles form a Star of David as depicted in the drawing. Prove that the shaded area is equal to the hatched area.

  

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  Topics:

  Geometry -> Area Calculation Proof and Example Set Theory Geometry -> Plane Geometry -> Triangles -> Triangle Congruence

• Pentagon

  In a convex pentagon `ABCDE`, the following holds: `AE=AD`, `AB=AC`, and `angle CAD=angle ABE + angle AEB`.

  Let `AM` be the median to the side `BE` in the triangle `ABE`. Prove that `AM` is half the length of the segment `CD`.

  Topics:

  Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Rotation
  Sources:
  • Tournament of Towns, 1983-1984, Fall, Practice Version, Grades 9-10 Question 2 Points 3