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To reach B (fig. 158) from O, one may travel 3 divisions along towards X- to the left, and then 4 divisions upwards. Accordingly the point B is fixed by the co-ordinates (-3, 4).

To reach C from O, go 3 divisions along to the right, then 4 divisions downwards. C is therefore (3, − 4).

N.B. To the right is reckoned +; to the left,

Upwards is reckoned +; downwards -.

To get from O to E, it is not necessary to travel along at all; the journey is simply 4 divisions upwards. Accordingly, E is the point (0, 4).

Ex. 813. Write down the co-ordinates of the following points in fig. 158: D, F, G, H, O, P, Q, R, S.

Ex. 814. Plot (i.e. mark on squared paper) the following points: (5,0), (4,3), (3, 4), (0,5), (−3, 4), (−4, 3), (−5, 0), (-4, −3), (−3, −4), (0, −5), (3, 4), (4, -3), (5, 0).

Ex. 815. Plot the points: (8, 16), (6, 9), (4, 4), (2, 1), (0, 0), (−2, 1), (-4, 4), (-6, 9), (−8, 16).

Ex. 816. Plot the points: (0, 0), (2, 0), (−2, 0), (0, 13), (1, −10), (8, 6), (-8, -6), (-3, -5). (The constellation of Orion.)

Ex. 817. Plot the points: (-12, -2), (-8, 0), (−4, 0), (0, 0), (3, −2), (7, 0), (5, 4). (The Great Bear.)

Ex. 818.

(Inch paper.) Find the co-ordinates of two points each of which is 3 inches from (0, 0) and (2, 2).

Ex. 819. (Inch paper.) Find the co-ordinates of all the points which are 2 inches from the origin and 1 inch from the x-axis (XOX).

Ex. 820. (Inch paper.) Find the co-ordinates of all the points which are equidistant from the two axes and 3 inches from the origin.

Ex. 821. (Inch paper.) Find the co-ordinates of a point which is equidistant from

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(ii) (2, 3), (2, −1),

(iii) (2, 3), (2, −1), (−2, − 2).

Ex. 822. (Inch paper.)

Find the co-ordinates of a point inside the triangle given in Ex. 821 (i), and equidistant from its three sides.

Ex. 823. Repeat Ex. 822 for the triangles given in Ex. 821 (ii) and (iii).

MISCELLANEOUS EXERCISES.

CONSTRUCTIONS.

Ex. 824. A ship is sailing due N. at 8 miles an hour. At 3 o'clock a lighthouse is observed to be N.E. and after 90 minutes it is observed to bear 71° S. of E. How far is the ship from the lighthouse at the second observation, and at what time (to the nearest minute) was the ship nearest to the lighthouse?

Ex. 825. Is it possible to make a pavement consisting of equal equilateral triangles?

Is it possible to do so with equal regular figures of (i) 4, (ii) 5, (iii) 6, (iv) 7 sides?

Ex. 826. A triangle ABC has ▲ B=60°, BC=8 cm.; what is the least possible size for the side CA? What is the greatest possible size for LC?

Ex. 827. Draw a triangle ABC and find points P, Q in AB, AC such that PQ is parallel to the base BC and BC. Give a proof.

[Trisect the base and draw a parallel to one of the sides.]

Ex. 828. In OX, OY find points A, B such that ▲OAB=34OBA. Give a proof.

[Make an angle equal to the sum of these angles.]

Ex. 829. A and B are two fixed points in two unlimited parallel straight lines; find points P and Q in these lines such that APBQ is a rhombus. Give a proof.

Ex. 830. Prove the following construction for bisecting the angle BAC :-With centre A describe two circles, one cutting AB, AC in D, E, and the other cutting them in F, G respectively; join DG, EF, intersecting in H ; join AH.

Ex. 831. A, B are two points on opposite sides of a straight line CD; in CD find a point P so that LAPC=4BPC. Give a proof. (See Ex. 572.)

Ex. 832. Construct a rhombus PQRS having its diagonal PR in a given straight line and its sides PQ, QR, RS passing through three given points L, M, N respectively. Give a proof.

Ex. 833. A and B are two given points on the same side of a straight line CD; find the point in CD the difference of whose distances from A and B is greatest.

Also find the point for which the difference is least.

Ex. 834: A and B are two points on the same side of a straight line CD; find the point P in CD for which AP + PB is least. Give a proof. (See Ex. 572.)

Ex. 835. Describe a rhombus having two of its sides along the sides AB, AC of a given triangle ABC and one vertex in the base of the triangle. Give a proof.

Ex. 836. Draw a straight line equal and parallel to a given straight line and having its ends on two given straight lines. Give a proof.

Ex. 837. To trisect a given angle.

Much time was devoted to this famous problem by the Greeks and the geometers of the Middle Ages; it has now been shown that it is impossible with only the aid of a pair of compasses and a straight edge (ungraduated).

In fig. 160, DE=the radius of the circle; prove that BDE-4ABC.

fig. 159.

E

B

A

fig. 160.

Fig. 159 shows a simple form of trisector; the instrument is opened until the angle between the rods corresponding to BA and BC can be made to coincide with the given angle; then the angle between the long rods (corresponding to D) is one-third of the given angle.

With a ruler, marked on its edge in two places, and a pair of compasses, it is possible to trisect an angle as follows:

Let ABC be the angle. With B as centre and radius=the distance between the two marks describe a circle cutting BC at C; place the ruler so that its edge passes through C and has one mark on AB produced, the other on the circle (this must be done by trial, a pin stuck through the paper at C will help); rule the line DEC, then D=14 ABC,

THEOREMS.

Ex. 838. The gable end of a house is in the form of a pentagon, of which the three angles at the ridge and eaves are equal to each other: show that each of these angles is equal to twice the angle of an equilateral triangle.

Ex. 839. If on the sides of an equilateral triangle three other equilateral triangles are described, show that the complete figure thus formed will be (i) a triangle, (ii) equilateral.

Ex. 840. Two isosceles triangles are on the same base: prove that the straight line joining their vertices bisects the base at right angles.

Ex. 841. Two triangles ABC, DCB stand on the same base BC and on the same side of it; prove that AD is parallel to BC if AB=DC and AC=DB.

Ex. 842. In the diagonal AC of a parallelogram ABCD points P, Q are taken such that AP=CQ; prove that BPDQ is a parallelogram.

Ex. 843. ABCD, ABXY are two parallelograms on the same base and on the same side of it. Prove that CDYX is a parallelogram.

Ex. 844. The diagonal AC of a parallelogram ABCD is produced to E, so that CE CA; through E, EF is drawn parallel to CB to meet DC produced in F. Prove that ABFC is a parallelogram.

Ex. 845. E, F, G, H are points in the sides AB, BC, CD, DA respectively of a parallelogram ABCD, such that AH=CF and AE=CG: show that EFGH is a parallelogram,

Ex. 846. C is the mid-point of AB; from A, B, C perpendiculars AX, BY, CZ are drawn to a given straight line. Prove that, if A and B are both on the same side of the line, AX+BY=2CZ.

What relation is there between AX, BY, CZ when A and B are on opposite sides of the line?

Ex. 847. If the bisectors of the base angles of an isosceles triangle ABC meet the opposite sides in E and F, EF is parallel to the base of the triangle.

Ex. 848. In a quadrilateral ABCD, AB = CD and ▲ B= ¿C; prove that AD is parallel to BC.

Ex. 849. Prove that the diagonals of an isosceles trapezium are equal,

Ex. 850. ABCD is a quadrilateral, such that ▲ A= ▲ B and ▲ C= ▲ D; prove that AD=BC.

Ex. 851. The figure formed by joining the mid-points of the sides of a rectangle is a rhombus.

Ex. 852. The medians BE, CF of a triangle ABC intersect at G; GB, GC are bisected at H, K respectively. Prove that HKEF is a parallelogram. Hence prove that G is a point of trisection of BE and CF.

Ex. 853. The diagonal AC of a parallelogram ABCD is produced to E, so that CE CA; through E and B, EF, BF are drawn parallel to CB, AC respectively. Prove that ABFC is a parallelogram.

Ex. 854. T, V are the mid-points of the opposite sides PQ, RS of a parallelogram PQRS. Prove that ST, QV trisect PR.

Ex. 855. Any straight line drawn from the vertex to the base of a triangle is bisected by the line joining the mid-points of the sides.

Ex. 856. The sides AB, AC of a triangle ABC are produced to X, Y respectively, so that BX=CY=BC; BY, CX intersect at Z. Prove that LBZX+4BAC=90°.

Ex. 857. ABCD is a parallelogram and AD=2AB; AB is produced both ways to E, F so that EA=AB=BF. Prove that CE, DF intersect at right angles.

Ex. 858. In a triangle whose angles are 90°, 60°, 30° the longest side is double the shortest.

[Complete an equilateral triangle.]

Ex. 859. In a right-angled triangle, the distance of the vertex from the mid-point of the hypotenuse is equal to half the hypotenuse.

[Join the mid-point of the hypotenuse to the mid-point of one of the sides.]

Ex. 860. Given in position the right angle of a right-angled triangle and the length of the hypotenuse, find the locus of the mid-point of the hypotenuse. (See Ex. 859.)

Ex. 861. ABCD is a square; from A lines are drawn to the midpoints of BC, CD; from C lines are drawn to the mid-points of DA, AB. Prove that these lines enclose a rhombus.

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