Modern Geometry

Convexity refers to a property of:

• Curved shapes

• Straight lines

• Concave shapes

• Both A and C

In geometry, a convex set is defined as a set where:

• Any two points can be connected by a straight-line segment within the set

• Any two points lie on the boundary of the set

• Any two points are equidistant from the center of the set

• Any two points form a concave shape within the set

Separation in convex sets refers to:

• The ability to divide a set into two non-overlapping parts

• The ability to connect two points within the set with a straight line

• The ability to form a concave shape within the set

• The ability to create a curved boundary for the set

The Hahn-Banach separation theorem is a fundamental result in: A) Convex optimization B) Algebraic geometry C) Non-Euclidean geometry D) Differential geometry

• Convex optimization

• Algebraic geometry

• Non-Euclidean geometry

• Differential geometry

What is the sum of the interior angles of a convex quadrilateral? A) 180 degrees B) 270 degrees C) 360 degrees D) 450 degrees

• 180 degrees

• 270 degrees

• 360 degrees

• 450 degrees

Which of the following is NOT a property of a convex quadrilateral? A) The sum of the lengths of any two sides is greater than the sum of the lengths of the other two sides. B) The diagonals are congruent. C) The opposite angles are supplementary. D) The opposite sides are parallel.

• The sum of the lengths of any two sides is greater than the sum of the lengths of the other two sides.

• The diagonals are congruent.

• The opposite angles are supplementary.

• The opposite sides are parallel.

The diagonals of a convex quadrilateral: A) Always lie inside the shape B) Are always equal in length C) Bisect each other D) All of the above

• Always lie inside the shape

• Are always equal in length

• Bisect each other

• All of the above

What is the area of a convex quadrilateral with diagonals of length 12 cm and 16 cm, and the angle between them is 60 degrees? A) 72 sq. cm B) 96 sq. cm C) 144 sq. cm D) 192 sq. cm

• 72 sq. cm

• 96 sq. cm

• 144 sq. cm

• 192 sq. cm

Which of the following is NOT a property of a cyclic quadrilateral? A) The sum of opposite angles is 180 degrees B) The diagonals are equal in length C) The vertices lie on the circumference of a circle D) The diagonals intersect inside the shape

• The sum of opposite angles is 180 degrees

• The diagonals are equal in length

• The vertices lie on the circumference of a circle

• The diagonals intersect inside the shape

What is angular measure? A) The measure of the angle between two lines B) The measure of the distance between two points C) The measure of the curvature of a circle D) The measure of the distance between two parallel lines

• The measure of the angle between two lines

• The measure of the distance between two points

• The measure of the curvature of a circle

• The measure of the distance between two parallel lines