PROBLEM 19.

To draw a parallelogram equal in area to a given rectilineal figure, and having an angle equal to a given angle.

Let ABCD be the given rectil. fig., and E the given angle. It is required to draw a parm equal to ABCD and having an angle equal to E.

Through C draw CF par1 to DB, and meeting AB produced

NOTE. If the given rectilineal figure has more than four sides, it must first be reduced, step by step, until it is replaced by an equivalent triangle.

EXERCISES.

(Reduction of a Rectilineal Figure to an equivalent Triangle.)

1. Draw a quadrilateral ABCD from the following data: AB BC=55 cm. ; CD=DA=4.5 cm. ; the LA=75°. Reduce the quadrilateral to a triangle of equal area. Measure the base and altitude of the triangle; and hence calculate the approximate area of the given figure.

2. Draw a quadrilateral ABCD having given :

AB=28", BC=3.2′′, CD=3′3′′, DA=3′6′′, and the diagonal BD=3′0′′. Construct an equivalent triangle; and hence find the approximate area of the quadrilateral.

3. On a base AB, 4 cm. in length, describe an equilateral pentagon (5 sides), having each of the angles at A and B 108°.

Reduce the figure to a triangle of equal area; and by measuring its base and altitude, calculate the approximate area of the pentagon.

4.

A quadrilateral field ABCD has the following measurements: AB=450 metres, BC=380 m., CD=330 m., AD=390 m., and the diagonal AC=660 m.

Draw a plan (scale 1 cm. to 50 metres). Reduce your plan to an equivalent triangle, and measure its base and altitude. Hence estimate the area of the field.

(Problems. State your construction, and give a theoretical proof.)

5. Reduce a triangle ABC to a triangle of equal area having its base BD of given length. (D lies in BC, or BC produced.)

6. Construct a triangle equal in area to a given triangle, and having a given altitude.

7. ABC is a given triangle, and X a given point. Draw a triangle equal in area to ABC, having its vertex at X, and its base in the same straight line as BC.

8. Construct a triangle equal in area to the quadrilateral ABCD, having its vertex at a given point X in DC, and its base in the same straight line as AB.

9. Shew how a triangle may be divided into n equal parts by straight lines drawn through one of its angular points.

10. Bisect a triangle by a straight line drawn through a given point

in one of its sides.

[Let ABC be the given ▲, and P the

given point in the side AB.

Bisect AB at Z; and join CZ, CP.
Through Z draw ZQ parallel to CP.
Join PQ.

11.

Trisect a triangle by straight lines drawn from a given point in one of its sides.

[Let ABC be the given A, and X the

given point in the side BC.

12. Cut off from a given triangle a fourth, fifth, sixth, or any part required by a straight line drawn from a given point in one of its sides.

13. Bisect a quadrilateral by a straight line drawn through an angular point.

[Reduce the quadrilateral to a triangle of equal area, and join the vertex to the middle point of the base.]

14. Cut off from a given quadrilateral a third, a fourth, a fifth, or any part required, by a straight line drawn through a given angular point.

AXES OF REFERENCE.

COORDINATES.

EXERCISES FOR SQUARED PAPER.

If we take two fixed straight lines XOX', YOY' cutting one another at right angles at O, the position of any point P with reference to these lines is known when we know its distances from each of them. Such lines are called axes of reference, XOX' being known as the axis of x, and YOY' as the axis of y. Their point of intersection O is called the origin.

X'

The lines XOX', YOY' are usually drawn horizontally and vertically.

In practice the distances of P from the axes are estimated thus:

From P, PM is drawn perpendicular to X'X; and OM and PM are measured.

OM is called the abscissa of the point P, and is denoted by x.

The abscissa and ordinate taken together are called the coordinates of the point P, and are denoted by (x, y).

We may thus find the position of a point if its coordinates are known.

EXAMPLE.

Plot the point whose coordinates are (5, 4).
Along OX mark off OM, 5 units in length.

At M draw MP perp. to OX, making MP 4 units in length.
Then P is the point whose coordinates are (5, 4).

The axes of reference divide the plane into four regions XOY, YOX', X'OY', Y'OX, known respectively as the first, second, third, and fourth quadrants.

It is clear that in each quadrant there is a point whose distances from the axes are equal to those of P in the above diagram, namely, 5 units and 4 units.

The coordinates of these points are distinguished by the use of the positive and negative signs, according to the following system:

Abscissæ measured along the x-axis to the right of the origin are positive, those measured to the left of the origin are negative. Ordinates which lie above the x-axis (that is, in the first and second quadrants) are positive; those which lie below the x-axis (that is, in the third and fourth quadrants) are negative

Thus the coordinates of the points Q, R, S are

(− 5, 4), (−5, −4), and (5, −4) respectively.

NOTE. The coordinates of the origin are (0, 0).

In practice it is convenient to use squared paper. Two intersecting lines should be chosen as axes, and slightly thickened to aid the eye, then one or more of the lengthdivisions may be taken as the linear unit. The paper used in the following examples is ruled to tenths of an inch.

EXAMPLE 1. The coordinates of the points A and B are (7, 8) and (-5, 3): plot the points and find the distance between them.