Answer all the questions below and press submit to see how many you got right.
Boundary : The boundary of a 2-dimensional shape is the 1-dimensional shape that separates the 2-dimensional shape from the rest of the 2-dimensional space.
Circle : A circle is a two-dimensional shape. The boundary of a circle is a set of points that has the same distance (the radius) from the center of the circle.
Constant : Constant is another word for a fixed number that is mainly used in the context of expressions or functions like $f(x)=c$ that are equal to the same number irrespective of any variable.
Ellipse : An ellipse is a two-dimensional shape with a boundary that is a set of points for which the combined distance from two points is constant. Circles are special cases of ellipses. The boundary of an ellipse can also written as the set of points $\{(x,y)| \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \}$
Point : A point is an element in a space. Shapes are made of sets of points.
Set : A set is a collection of objects.
Choose all statements that are true for an ellipse.
Addition : Addition is the mathematical operation that describes increasing a number by an amount equal to a second number. The mathematical symbol for addition is the plus sign $+.$ The term addition is also used for a generalization of this basic operation on numbers to functions, vectors and matrices.
Angle : If two line segments (or rays) both start at a common point the opening between the two line segments is called an angle. The common point is called vertex of the angle. The size of an angle is measured in degrees.
Area : The amount of unit squares that is needed to cover a 2-dimensional shape.
Center : The center of a circle is the unique point in the circle, such that all points on the boundary of the circle have a constant distance equal to the radius from that point. The center of a regular polygon is the point that has equal distance from all its corners.
Counterexample : A counterexample to a statement of the form 'all objects of type x have property y' is an object of type x that does not have property y. A counterexample to a statement shows that the statement is not true in general. For example the number 3 is a counterexample to the statement that all numbers are even.
Height : In a triangle a height (or altitude) is the shortest line segment that connects a side to its opposing corner. A height of a triangle is perpendicular to the side is started at. The three heights of a triangle intersect in a point, the so called orthocenter of the triangle. In a parallelogram the height is the shortest distance between two opposing sides. In a pyramid the height is the shortest distance between the base area and the corner of the pyramid that is not part of the base area. In a prism the height is the shortest distance between the two base faces.
Independence : Random variables $X,$ $Y$ are called independent if $P[X\in A, Y\in B]=P[X \in A]P[Y \in B].$ Independent identically distributed random variables feature prominently in the law of large numbers and the central limit theorem. Vectors $x_1,x_2\ldots, x_n$ are called linearly independent if $\lambda_1 x_1+\lambda_2 x_2+\ldots+\lambda_n x_n=0$ implies $\lambda_1=\lambda_2=\ldots=\lambda_n=0.$
Line : A line AB is a one-dimensional shape that includes the points A and B, all the points on the line segment in between A and B and all the points of the straight extension of the line segment beyond A and B. A line does not have an endpoint.
Perpendicular line : Two lines are called perpendicular if they cross in a right angle.
Right angle : A right angle is an angle equal to $90^{\circ}.$
Thales' theorem : Given a circle, points A, B and C on the circle such that the line segment AB goes through the center of the circle, the theorem of Thales says that the angle ACB is a right angle making ABC a right triangle. Thales' theorem is a special case of the inscribed circle theorem.
Theorem : A mathematical result that has been proven to hold true under the assumptions that are stated in the theorem. The most famous theorems have name like for example the Pythagorean theorem or Fermat's little theorem.
Triangle : A triangle is a polygon with three corners and three sides. You can calculate the area of a triangle by multiplying half the length of the base by the height on that base. The sum of the interior angles in a triangle is always $180^{\circ}.$
A line crosses the center of a circle and crosses the circle in points A and B. We pick any point C on the circle not equal to A or B. Choose all the statements about the triangle ABC that are true.
Length : Length is the attribute of a one-dimensional shape that can be measured with a measuring tape.
Line segment : A line segment AB is a one-dimensional shape consisting of the points A and B and all the points on the straight connection of A and B. A and B are both endpoints of the line segment AB.
Parallel : Two lines in a two-dimensional plane are called parallel if they never cross.
Ratio : A ratio is a comparison of two numbers using a division.
Similar triangle : A similar triangle is a triangle with the same shape but not necessarily the same size. Ratios of corresponding side lengths and corresponding angles are the same in similar triangle. This fact is used to be able to uniquely define the trigonometric functions using ratios of side lengths in right triangles. Two triangles are similar if they share two angles (and therefore all three) or if all 3 ratios of the lengths of corresponding sides in the two triangles are the same. Two triangles are also similar if two ratios of lengths of corresponding sides in the two triangles are the same and the angle between the two sides in the two triangles is the same.
A line PQ and a line XY cross each other in a point O. Choose all statements that are true if the line PX is parallel to the line QY.