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1022"

11

12"

Problem XII.-To construct a quadrilateral given the four sides and one angle.

Construct a quadrilateral ABCD given AB=1 in., BC = 1.5 in., CD = 1.2 in., DA = 1.2 in., B= 75°. [See Fig. 21. We can construct A ABC

D by the method of Problem VI., as we know

A А. two sides and the included angle; we can then construct AADC by the method of Problem I.]

Draw ZĀBC= 75°, and mark off on its arms lengths BA = 1" and BC=1:5".

B 1.5"

C With centres A and C and radii 1.2"

Fig. 21. describe arcs intersecting at D.

The student will be able to solve many other simple cases by first drawing a rough sketch, and finding from it how the figure can be built up from the given data.

175°

20 mm.,

Examples VII. Construct quadrilaterals ABCD from the following data :1. AB = BC = 1", CD = •3", B = 90°, C = 90; measure AD. 2. AB = 1.5", BC = 1", CD : 1", DA = 2", AC = 2"; measure BD. 3. AB 5 cm., BC = 3 cm., CD =6 cm., DA=1 cm., ABC=90°; measure ZA. 4. DA

AB 40 mm., BC=40 mm., A= 60°, B=120°; measure 2C. 5. AB = 30 mm., BC = 35 mm., CD 25 mm., DA

BD 45 mm.; measure LB.

6. DA = 2.1", AB = 1.4", BC = 1", CD = 1", A = 50°; measure AC. 7. AB = 2", BC = 1:5", CD = 2.2", AC = 2.2", BD = 2:2", measure AD. 8. BC 40 mm., ABC = 80°, DBC = 35°, DCB = 70°, ACB=40°; measure AD.

9. BD 32 mm., DA = 32 mm., BC = 20 mm., ADB = 30°, DBC = 25°, measure AC.

30 mm.,

11. On parallels.-Learn Definition 30 on page 6. The meaning of the word parallel should be easy to understand. The lines drawn on an ordinary sheet of ruled paper form a set of parallel lines; no two of them meet, or would meet even if produced beyond the paper.

Parallel lines point in the same direction. Thus if two lines are parallel and one of them points due north, the other will also point due north.

Look at the parallel lines in various figures in the book. On page 61, AB is parallel to CD, on page 64, AB, CD, and PQ are all parallel; on page 66, BA is parallel to CE, and on page 73, we have AD parallel to BC, and AB parallel to DC.

In Definition 30 page 6, notice the words “ in the same plane." The word plane simply means a flat surface ;-either a real flat surface, such as the top of a table, or an imaginary flat surface such as we can conceive stretched across between the posts of the goal at football.

Now it is quite possible to draw two lines which do not meet when produced, and yet which are not parallel. For instance draw a line on the table pointing north and south, and a line on the floor pointing east and west. These two lines would not meet when produced, and yet are not parallel, for they point in quite different directions. But these lines are not in the same plane, for you could not imagine a flat surface which contained both lines. Thus we cannot assert that two lines are parallel unless (i) they would never meet, and (ii) they lie in the same plane.

We shall frequently use the symbol || for parallel.

Problem XIII.To draw a line parallel to a given line at a given distance, Draw a line || to AB at a distance of .8".

с

D Take any two points in the line H and K. With centres H and K and radius •8" describe two arcs on the same side of

A H the line.

K B Draw CD touching each of these arcs (as shown in the figure). CD Fig. 22. will be the required parallel.

Every point in the line CD will be at a distance •8" from the line AB. Note that the distance of a point from a line is the length of the perpendicular drawn from the point to the line.

E

H Problem XIV.-To draw a line through

F a given point parallel to a given line,

Draw through H a line | to AB (Fig. 23).

А с
A G

DB Take any convenient point C in AB.

Fig. 23.
With centre C and radius CH draw
the arc HD, cutting AB in D. With
centre H and radius HC draw H-、
the

Р
CE. From CE cut off
CF DH (measuring with the

M compasses). FH is the required A

B parallel

Alternative Method.-Place one set square so that one side lies along AB, as in the position MCD (Fig. 24). Place the other set square against

Fig. 24.

arc

it, in the position EFG. Keeping EFG fixed, slide the first set square along it till the side CB reaches the point P, as in the position HKL. Rule the line HPK, which will be the required parallel.

We may also draw the parallel by using parallel rulers.

Examples VIII.

1. Take a line AB of length 2.5". Bisect it at C. Draw lines I to AB at the points A, C, and B respectively. Note that these lines are parallel. Join any point H on the first perpendicular to any point k on the third, and show by measurement that HK is bisected by the second perpendicular.

2. Take a line AB of length 2:4". Bisect it at C. Construct an angle BAD 40°. Through C draw a line CE || to AD. From B draw a line I to AD meeting AD in F and CE in G. Measure the angles BCE and BGE, and the lengths BG, GF.

3. Draw three parallel lines. Draw a line cutting all three lines and making an angle of 50° with one of them. Show that every angle in the figure measures either 50° or 130°.

4. Draw a line, a second parallel to it, and 1 inch above; a third parallel to the second, and 2 inches above. Draw a line making an angle of 42° with the lowest line, and cutting the other lines. Measure all the angles in the figure.

5. Draw a line, and two other lines parallel to it. Draw a line at right angles to one of the lines and cutting the other lines, and measure all the angles in the figure.

6. Draw AB of length 1.5". Make an angle BAC = 35°, and an angle ABD 35° (AC and BD being on opposite sides of AD). Note that AC is || to BD. Make AC = BD = 1.7". Join CD, cutting AB in 0.

Measure AO, BO, CO, DO. Note that AB and CD bisect one another.

7. Construct an equilateral triangle of side 4 cm. Through the vertex draw a line || to the base. Measure the distance of any point on this parallel from the base.

8. Draw a triangle having sides 30, 40, 50 mm. Draw a line || to the longest side through the opposite vertex of the triangle. Measure the distance between these parallels. Learn these rules :(i) If a line is perpendicular to one of a set of parallel

lines, it is perpendicular to them all. (Compare question 5.) (ii) If a line cut two or more parallel lines, and is not perpen

dicular to them, all the acute angles formed are equal, and all the obtuse angles formed are equal, and any acute angle

together with any obtuse angle makes up 180°. (Compare question 4.)

12. On parallelograms.—Learn Definitions 31-34 on page 7, and Definition 1 on page 113. Note that all the angles of a rectangle are right angles.

Problem XV.—To construct a parallelogram given the two sides and the included angle. (See figure on page 72.)

Draw an angle ACD of the given magnitude; and make the arms CD and AC of the given lengths. Through A draw a line ll to CD, and through D drawn a line || to ĂC. These lines complete the parallelogram. (The diagonal CB is not wanted.)

A similar construction enables us to describe a rectangle, square, or rhombus, as in all these figures opposite sides are parallel.

Examples IX. 1. Draw a rectangle whose sides measure and 4 cm., and measure both diagonals.

2. Draw a rhombus each of whose sides is 2" and whose acute angle is 60°. Measure the diagonals and show that they bisect one another at right angles.

3. Draw a parallelogram on a base 2", having the left angle 50° and the adjacent side = 1". Show by measurement that opposite angles are equal.

4. On base 1:5" describe above it an isosceles triangle, having each side 2", and another isosceles triangle below it having each side 2.5", and measure the line joining the vertices of the two triangles. Also show by measurement that this line bisects the base.

5. Draw a square of side 2" and measure each diagonal. Show that the diagonals bisect one another.

6. Draw any two straight lines AB and CD which bisect one another at a point 0, at any angle. Join AC, CB, BD, and DA. Show, by measurement, that the two angles DAC and ACB together measure 180°.

7. In the last example if AB were 6 cm. and CD 4 cm. and the angle COA 45°, measure all the other lines and angles.

8. On AB 2" long as diagonal draw a parallelogram whose side BC is 1.5" long and whose side AC is 2.5", and measure the angles of the parallelogram and those at the intersection of the diagonals.

9. Construct the quadrilateral ABCD whose sides are l", 1", 1.5", and l", and one of whose diagonals is 1.7". Measure the other diagonal.

10. Construct the quadrilateral ABCD. AB (1") parallel to DC (2") at a distance of 1" from one another, the angle BAD being 120°; measure AC.

11. Construct a quadrilateral ABCD. AB (1.5") parallel to CD (2"). AD being 2", and the diagonal AC 3". Measure the angles.

12. Construct a rectangle ABCD, given diagonal AC 2", and 2 ACD = 40°. Measure its sides.

13. Draw two lines of lengths 1.4" and 2:4" bisecting each other at right angles. Measure the sides of the quadrilateral of which these lines are diagonals,

14. Draw a circle of radius 1.5". Draw two diameters cutting at an angle of 40°. Measure the angles of the quadrilateral of which these are the diagonals.

15. Draw a circle of radius 1.4". Draw two diameters at right angles. Measure the sides and angles of the quadrilateral of which these are the diagonals.

16. Describe a rectangle in which a diagonal is 2:4", and one side is 1.2". Measure the longer side. (Start with the given side.)

17. Knowing that the diagonals of a parallelogram bisect each other, construct a parallelogram whose longer side is 35 mm., and whose diagonals are 50 and 40 mm. respectively. Measure the other sides.

18. Knowing that the diagonals of a square are equal and bisect each other at right angles, construct a square of diagonal 2.8 inches. Measure its sides.

ANSWERS TO THE EXAMPLES IN THE INTRODUCTORY COURSE.

II. 1. EDF (or FDE).

2. DEF (or FED). 3. PQ and PR. 4. YX and YZ. 5. POR (or RQP). 6. ZYX (or XYZ).

7. PQR. 8. QPR. 9. XZY. 10. Greater.

III. 1. (a) Twice, (6) four times, (c) six times, (d) ten times. 2. PG. 3. PF. 4. PF. 5. Twice. IV. 1. 1.225". 2. .75", 1.25". 3. 1.79".

4. 23.6 mm. 5. 46.8 mm.

6. 1.73", .67". V. 1. 751°, 751°, 29o. 2. 90°, 53°, 37o. 3. 117°, 37°, 26°. 4. 120°, 22°, 38°.

5. 60°, 60°, 60°. 6. 90°, 673, 221°. 7. 120°, 34°, 26°. VI. 1. 1 in., 1.73 in., 90°.

2. 2:45 in., 3.35 in., 60°. 3. 1.41 in., 1 in., 75o.

4. 1.01 in., 1.56 in., 100°. 5. 25 mm., 25 mm., 60°.

6. 25.7 mm., 32.9 mm., 60°. 7. BC = 42.4 mm., B = C = 45°. 8. AB = 20 mm., AC = 34.6 mm., B = 60°. 9. B = 45°, AC 24.5 mm., AB 9 mm. 10. AB = 3.02 in., B = 39°, C = 719. 11. BC = 18.5 mm., B = C = 72o. 12. BC = 29 mm.,

A 25°, B 35°. 13. AB 66", B = 134°, C 11°, or AB = 3·44", B = 46°, C = 99o. 14. BC 17.8 mm., A = 35°, B 105°, or BC = 28.2 mm., A = 65°, B = 75°. 15. 15 mm. 16. 18.9 mm.

17. AC = 3.78 cm., C = 34°. 18. AC = 3.46 cm., A = 90°, C = 30°.

VII. 1. 1•22". 2. 1.75". 3, 126o. 4. 41°. 5. 75o. 6. 2.13".

7. 35".
8. 27.6 mm.

9. 28.1 mm. VIII. 2. 40°, 90°, 77", -77". 4. They are all either 42° or 138o. 5. All right angles. 6. .75", .75", 1.16", 1•16". 7. 34.6 mm. 8. 24 mm. IX. 1, 5 cm. each.

2. 2", 3.46". 3. There are two angles of 50° and two of 130°. 4. 4.24". 5. Each 2.83". 17. Two sides measure 4.63", and the other two 2-12". Two angles measure

591°, and the other two 120.1°. 8. Angles at C and D each 53o; at A and B each 127°; angles at the inter

section of diagonals are 56° and 124°. 9. 1.45". 10. 1.74". 11. A, 83°; B, 111°; C, 69°; D, 97°. 12. 1.29", 1:53". 13. Each 1:39". 14. Each 90°. 15. Each side 1.98", each angle 90°. 16. 2.08". 17. 28.7 mm. 18. 1.98".

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