Diagonal connects non-adjacent tops of a polygon which number of the parties not less than four. Calculate this size through initial or intermediate these tasks, using the corresponding formulas.

## Instruction

1. In any closed geometrical figure consisting not less than of four pieces it is possible to carry out at least two diagonals. So many diagonals can have a quadrangle: parallelogram, rectangle, rhombus and square.

2. Find parallelogram diagonals if it is known that one of them is more another on 1, and lengths of the parties are equal to a=5 and b=7. In this respect in geometry there is a ready formula according to which the sum of squares of lengths of diagonals is equal to the doubled sum of squares of the parties: d1² + d2² = 2 • (a² + b²) = 2 • (25 + 49) = 148.

3. Substitute d2=d1+1: d1² + (d1+1)² = 148 2•d1² + 2•d1 + 1 = 148.

4. Solve the following equation of rather unknown d1:2 • d1² + 2•d1 – 147 = 0D = 4 + 4•2•147 = 1180d1 = (-2 + √1180)/4 ≈ 8.1 → d2 = 9.1.

5. The formula for a rectangle becomes simpler as its diagonals are equal among themselves: 2 · d² = 2 • (a² + b²) = 2 • (25 + 49) = 148 → d² = 74 → d ≈ 8.6.

6. In case of a square the situation is even more simply, its diagonals not only have equal length, but also are directly proportional to the party: 2 · d² = 4•a² → d² = 2•a² → d = √2•a = [a=5] = √2•5 ≈ 7.

7. A rhombus – a special case of a parallelogram with the equal parties, however unlike a square of diagonal are not equal among themselves. Let's assume that the party of a rhombus of a=5, and length of one of diagonals is equal to 3. Then: d1² + 9 = 4•25 → d1 = 9.

8. Diagonals can be carried out not only in a flat figure, but also in spatial. For example, in a parallelepiped. The square of length of diagonal of a rectangular parallelepiped (or its special case - a cube) is equal to the total size of squares of three of its dimensions. Measurements are called the edges having one general top.

9. Diagonals has no triangle and its three-dimensional option – a tetrahedron as they have no non-adjacent tops. The number of diagonals in any n-polygon can be determined as follows: nd = (n² – 3•n)/2.