This fact makes it possible to introduce a large number of classical structures which induce such invariants of algebraic varieties that can be obtained by purely algebraic means only with great difficulty or perhaps not all. In the case of algebraic geometry over the field of complex numbers, every algebraic variety is simultaneously a complex-analytic, differentiable and topological space in the ordinary Hausdorff topology. In turn, algebra provides a flexible and powerful apparatus which is just as suitable for the conversion of a tentative reasoning into a proof as for the formulation of such proofs in their most obvious and most general form. This language implies problems, constructions and considerations which do not follow from the point of view of pure algebra. If the coordinates are real numbers, and if the space is two- or three-dimensional, the situation may be clearly visualized however, the language of geometry is also used in more general situations. The geometrical intuition appears when every 'set of solutions' is identified with a 'set of points in a coordinate space'. Scheme Algebraic space).Īlgebraic geometry may be 'naively' defined as the study of solutions of algebraic equations. Algebraic variety) and their various generalizations (schemes, algebraic spaces, etc., cf.

One piece is 2 m longer than the other.The branch of mathematics dealing with geometric objects connected with commutative rings: algebraic varieties (cf.

An electrician cuts a 30 m piece of cable into two pieces.

A lab technician cuts a 12 cm piece of tubing into two pieces such that one piece is two times longer than the other.

If the perimeter of a rectangle is 280 m and the width is 26 m less than the length, find its length and width.

Find the length and width of a rectangular garden if the perimeter is 152 m and the width is 22 m less than the length.

If the perimeter of a rectangle is 304 cm and the length is 40 cm longer than the width, find the length and width.

Find the dimensions of a rectangle if the perimeter is 150 cm and the length is 15 cm greater than the width.

The measure of the third angle is 20° greater than the first.

The second angle of a triangle is twice as large as the first.

The third angle is three times as large as the first.

Two angles of a triangle are the same size.

The third angle is 12° smaller than the first angle. The third angle is 12° larger than the first angle.

The second angle of a triangle is the same size as the first angle.

A 48 m piece of hose is to be cut into two pieces such that the second piece is 5 m longer than the first.įor questions 9 to 18, write and solve the equation describing each relationship.

The first piece is five times as long as the second.

A 140 cm cable is cut into two pieces.

The sum of the first and second angles of a triangle is half the amount of the third angle.

The first angle of a triangle is half as large as the second and 20° larger than the third.

The first angle of a triangle is twice as large as the second and 10° larger than the third.

The length of a rectangle is 4 cm more than double its width, and the perimeter is 32 cm.The length of a rectangle is 8 cm less than double its width, and the perimeter is 64 cm.The length of a rectangle is 3 cm less than double the width, and the perimeter is 54 cm.This means that P_2 = 3 m and P_1 = 4 (3), or 12 m.įor questions 1 to 8, write the formula defining each relation. When looking at interior angles, the sum of the angles of any polygon can be found by taking the number of sides, subtracting 2, and then multiplying the result by 180°. It is common to run into geometry-based word problems that look at either the interior angles, perimeter, or area of shapes.

Algebraic geometry examples