# Algebra,trigonometry,plane geometry, coordinates and matrices

I would like to receive the all steps till the end of the each exercise and a written explanation or formulas

1. ABDC is a timber frame for a concrete base. It is shown to be parallelogram in shape: with ABCD12.000m; ACBD9.000m. It should be a rectangle.
How far, in the x direction, does corner A (or corner B) of the frame have to be moved to create a rectangle shape. Calculate to a precision of 1 mm.

2. The closed shape ABCD has the following data.

The angle , the coordinates of Point A are 2500.00mE, 1000.000mN

The internal angles of the loop are:

The distances are as follows:

Calculate the coordinates of the points B,C and D

3. Given the following system of three equations

Find the values of x, y, and z by
i) Crameras rule
ii) Matrix inversion

4. Angle and distance measurements have been made to determine the position of points, as in the diagram. The bearing of the line OP is . The observed internal angles and distances are as shown in the diagram. Calculate the area of OPQRS in square metres to a precision of 3 decimal places. The coordinates of Point O can be taken as (0.000mE, 0.000mN)

5. Given a regular hexagon PQRSTU where the coordinates of Point P are (15.615, 0.000), and point U are (41.297,-12.623).
Rotate the regular hexagon, about point P, such that both points P and Q lie on the x axis, with point U to the right of point P. Calculate the final coordinates for P,Q,R,S,T and U to a precision of 3 decimal places.

3. The equation of the offset circle whose centre is at point Oa, is given as .

Calculate to a precision of 1mm, the length of the arc AB.

As an aid to calculating, by reducing the size of the numbers involved, the dimensions of the x and y axes are in km. So A(-8366.000mE,3684.000mN) and B(-3000.000mE,7000.000mN)are the coordinates.

6. AAa and BBa represent two straight tangential sections of a road linked by a smooth curve AB. Curve AB is the arc of a circle centre Oa: an offset centred circle whose equation is given by .

The bearing of line BBa is , and that of AAa is

Calculate the length of the arc AB, to a precision of 1mm.

here necessary .