Ex. 771. Prove that any point on the bisector of an angle is equidistant from the arms of that angle.

Ex. 772. Prove formally that the locus of points at a distance of 1 inch from a given line, on one side of it, is a parallel line. (Take two such points, and show that the line joining them is parallel to the given line.)

Ex. 773. O is a fixed point. P moves along a fixed line; Q is in OP produced, and PQ=OP. Prove that the locus of Q is a parallel line.

INTERSECTION OF LOCI.

Draw, two unlimited straight lines AOA', BOB', intersecting at an angle of 45°. It is required to find a point (or points) distant 1 inch from each line.

First draw the locus of points distant 1 inch from AOA'; this consists of a pair of lines parallel to AOA' and distant 1 inch from it. The points we are in search of must certainly lie somethis locus.

where upon

Next draw the locus of points distant 1 inch from BOB'. The required points must lie upon this locus also.

The two loci will be found to intersect in four points. These are the points required.

Measure the distance from O of these points.

Ex. 774. Draw two unlimited straight lines intersecting at an angle of 80°. Find a point (or points) distant 2 cm. from the one line and 4 cm. from the other.

Ex. 775. Draw an unlimited straight line and mark a point O 2 inches from the line. Find a point (or points) 3 inches from O and 3 inches from the line. (What is the locus of points 3 inches from O? What is the locus of points 3 inches from the line? Draw these loci.) Measure the distance between the two points found.

Ex. 776. In Ex. 775 find two points distant 4 inches from O and from the line. Measure the distance between them.

Ex. 777. In Ex. 775 find as many points as you can distant 1 inch from both point and line.

Ex. 778. Given two points A, B 3 inches apart, find a point (or points) distant 4 inches from A and 5 inches from B.

Ex. 779. Make an angle of 45°; on one of the arms mark a point A 3 inches from the vertex of the angle.. Find a point (or points) equidistant from the arms of the angle, and 2 inches from A. Measure distance between the two points found.

Ex. 780. Draw a circle of radius 5 cm. and mark a point A 7 cm. from centre of circle. Find two points on the circle 3 cm. from A, and measure the distance between them.

Ex. 781. Construct

a quadrilateral ABCD, having AB=6 cm.,

BC 13 cm., CD=10 cm., 4ABC=70°, 4 BCD=60°.

On diagonal BD (produced if necessary), find a point (1) equidistant from A and C,

(2) equidistant from AB and AD,

(3) equidistant from AB and DC.

In each case measure the equal distances.

Ex. 782. Find two points on the base of an equilateral triangle (side 3 inches) distant 2.7 inches from the vertex. Measure distance between them.

Ex. 783. Find a point on the base of an equilateral triangle (side 10 cm.) which is 4 cm. from one side. Measure the two parts into which it divides

the base.

Ex. 784.

On the side AB of an isosceles triangle ABC (base BC=2 ins., ▲ A=36°), find a point P equidistant from the base and the other side AC. Measure AP, and the equal distances.

Ex. 785. In Ex. 784 prove that AP=CP=CB.

Ex. 786. Find a point on the base of a scalene triangle equidistant from the two sides. Is this the middle point of the base?

Ex. 787. Draw a circle of radius 2 ins.; a diameter; and a parallel line at a distance of 3 ins. Find a point (or points) in the circle equidistant from the two lines. Measure distance between these points.

Ex. 788. Draw a circle, a diameter AB, and a chord AC through A. Find a point P on the circle equidistant from AB and AC. Measure PB and PC.

Ex. 789. In Ex. 788, find a point on the circle equidistant from AB and CA produced.

Ex. 790. Draw A ABC having AB = 2.8 ins., AC=4.6 ins., BC=4.6 ins. Find a point (or points) equidistant from AB and AC, and 1 inch from BC. Measure distance between points.

Ex. 791. Using the triangle of Ex. 790, find a point (or points) equidistant from AB and AC, and also equidistant from B and C. Test the equidistance by measurement.

Ex. 792. In triangle of Ex. 790, find a point (or points) 2 inches from A, and equidistant from B, C. Measure the distance between them.

Ex. 793. Draw a triangle ABC; find a point O which is equidistant from B, C; and also equidistant from C, A. Test by drawing circle with centre O to pass through A, B, C.

Ex. 794. Two lines XOX', YOY' intersect at O, making an angle of 25°. A lies on OX, and OA=7 cm. Through A is drawn AB parallel to YOY'. Find a point (or points) equidistant from XOX' and YOY'; and also equidistant from AB and YOY'. Draw the equal distances and measure them.

Ex. 795. Draw a triangle ABC. Inside the triangle find a point P which is equidistant from AB and BC; and also equidistant from BC and CA. From P draw perpendiculars to the three sides; with P as centre and one of the perpendiculars as radius draw a circle.

Ex. 796. A river with straight banks is crossed, slantwise, by a straight weir. Draw a figure representing the position of a boat which finds itself at the same distance from the weir and the two banks.

Ex. 797. P is a moving point on a fixed line AB; O is a fixed point outside the line. P is joined to O, and PO is produced to Q so that OQ=PO. Prove that the locus of Q is a line parallel to AB. (See Ex. 772.)

Ex. 798.

Use the locus of Ex. 797 to solve the following problem. O is

a point in the angle formed by two lines AB, AC. Through O draw a line, terminated by AB, AC, and bisected at O.

Ex. 799. Draw a figure like fig. 157, making radius of circle 2 ins., CO=3 ins., CN=5 ins. Through O draw a line (or lines), terminated by AB and the circle, and bisected at O. (See Ex. 797.)

с

A

N

B

fig. 157.

A town X is 2 miles from a straight railway; but the two

Ex. 800.

the two stations.

stations nearest to X are each 3 miles from X. Find the distance between

CONSTRUCTION OF TRIANGLES, ETC. BY MEANS OF LOCI.

Ex. 801. Construct AABC, given

(i) base BC=14 cm., height=9 cm.,

B=65°. Measure AB.

(ii) AB=59 mm., AC=88 mm., height AD=49 mm. (Draw height first.) Measure base BC.

(iii) BC=4 in., ▲ B=80°, median CN=4 in. Measure BA.

(iv) base BC=12 cm., height AD=4 cm., median AL=5 cm. Measure AB, AC.

Ex. 802. Construct a right-angled triangle, given

(i) longest side=10 cm., another side=5 cm. Measure the smallest angle.

(ii) side opposite right angle = 4 in., another side 3 inches. Measure the third side.

Revise Ex. 556, 557.

Ex. 803. Construct a right-angled triangle ABC, given ▲ A=90°, AB=7 cm., distance of A from BC = 2.5 cm. Measure the smallest angle. Ex. 804. Construct an isosceles triangle having each of the equal sides twice the height. Measure the vertical angle.

Ex. 805.

Construct a triangle, given height = 2 in., angles at the extremities of the base = 40° and 60°. Find length of base.

Ex. 806.

Construct an isosceles triangle, given the height and the angle

at the vertex (without protractor).

Ex. 807. Construct a parallelogram ABCD, given

AB=12 cm., AD=10 cm., distance between AB, DC=8 cm. Measure the acute angle.

Ex. 808. Construct a rhombus, given that the distance between the parallel sides is half the length of a side. Measure the acute angle.

Ex. 809. Construct a quadrilateral ABCD, given diagonal AC=9 cm., diagonal BD = 10 cm., distances of B, D from AC 5 cm. and 4 cm. respectively, side AB=7 cm. Measure CD.

Ex. 810. Construct a trapezium ABCD, given base AB=10 cm., height=4 cm., AD = 4.5 cm., BC 4.2 cm. Measure angles A and B. (There are 4 cases.)

Ex. 811. Construct a trapezium ABCD, given base AB=3.5 in., height=1.7 in., diagonals AC, BD 2.5, 3.5 ins. respectively. Measure CD.

CO-ORDINATES.

Take a piece of squared paper; near the middle draw two straight lines intersecting at right angles (XOX, YOY in fig. 158). These will be called axes; the point O where they intersect will be called the origin.

In order to arrive at the point A, starting from the origin 0, one may travel 3 divisions along towards X+, to the right, and then 4 divisions upwards. Accordingly the point A is fixed These two numbers are called the

by the two numbers (3, 4). co-ordinates of the point A.

Ex. 812. Mark on a sheet of squared paper

(i) the points (3, 5), (3, 10), (8, 10), (8, 5).

(ii) the points (1, 2), (2, 4), (3, 6), (4, 8), (5, 10).

(iii) the points (4, 3), (4, 2), (4, 1), (4, 0), (4, −1), (4, −2).

(iv) the points (6, 6), (4, 6), (2, 6), (0, 6), (−2, 6).