し

EXERCISES ON PARALLELS AND PARALLELOGRAMS.

1. The straight line drawn through the middle point of a side of a triangle, parallel to the base, bisects the remaining side.

3. The straight line which joins the middle points of two sides of a triangle is equal to half the third side.

4. Shew that the three straight lines which join_the_middle points of the sides of a triangle, divide it into four triangles which are identi eally equal.

5. Any straight line drawn from the vertex of a triangle to the base is bisected by the straight line which joins the middle points of the other sides of the triangle.

6. ABCD is a parallelogram, and X, Y are the middle points of the opposite sides AD, BC: shew that BX and DY trisect the diagonal AC.

7 If the middle points of adjacent sides of any quadrilateral are joined, the figure thus formed is a parallelogram.

8. Shew that the straight lines which join the middle points of opposite sides of a quadrilateral, bisect one another.

9. From two points A and B, and from O the mid-point between them, perpendiculars AP, BQ, OX are drawn to a straight line CD. If AP, BQ measure respectively 4.2 cm. and 5.8 cm., deduce the length of OX, and verify your result by measurement.

Shew that OX=}(AP+BQ) or (AP BQ), according as A and B are on the same side, or on opposite sides of CD.

10. When three parallels cut off equal intercepts from two transversals, shew that of the three parallel lengths between the two transversals the middle one is the Arithmetic Mean of the other two.

11. The parallel sides of a trapezium are a centimetres and b centimetres in length. Prove that the line joining the middle points of the oblique sides is parallel to the parallel sides, and that its length is (a+b) centimetres.

12. OX and OY are two straight lines, and along OX five points 1, 2, 3, 4, 5 are marked at equal distances. Through these points parallels are drawn in any direction to meet OY. Measure the lengths of these parallels take their average, and compare it with the length of the third parallel. Prove geometrically that the 3rd parallel is the mean of all five.

State the corresponding theorem for any odd number (2n+1) of parallels so drawn.

13. From the angular points of a parallelogram perpendiculars are drawn to any straight line which is outside the parallelogram: shew that the sum of the perpendiculars drawn from one pair of opposite angles is equal to the sum of those drawn from the other pair. [Draw the diagonals, and from their point of intersection suppose a perpendicular drawn to the given straight line.]

14.

The sum of the perpendiculars drawn from any point in the base of an isosceles triangle to the equal sides is equal to the perpendicular drawn from either extremity of the base to the opposite side.

[It follows that the sum of the distances of any point in the base of an isosceles triangle from the equal sides is constant, that is, the same whatever point in the base is taken.]

How would this property be modified if the given point were taken in the base produced?

15. The sum of the perpendiculars drawn from any point within an equilateral triangle to the three sides is equal to the perpendicular drawn from any one of the angular points to the opposite side, and is therefore constant.

16. Equal and parallel lines have equal projections on any other straight line.

H.S.G.

DIAGONAL SCALES.

Diagonal scales form an important application of Theorem 22 We shall illustrate their construction and use by describing a Decimal Diagonal Scale to shew Inches, Tenths, and Hundredths.

A straight line AB is divided (from A) into inches, and the points of division marked 0, 1, 2.... The primary division OA is subdivided into tenths, these secondary divisions being numbered (from 0) 1, 2, 3, 9. We may now read on AB inches and tenths of an inch.

In order to read hundredths, ten lines are taken at any equal intervals parallel to AB; and perpendiculars are drawn through 0, 1, 2, ......

The primary (or inch) division corresponding to OA on the tenth parallel is now subdivided into ten equal parts; and diagonal lines are drawn, as in the diagram,

joining 0 to the first point of subdivision on the 10th parallel. 1 to the second

2 to the third

and so on.

The scale is now complete, and its use is shewn in the following example.

Example. To take from the scale a length of 2·47 inches.

(i) Place one point of the dividers at 2 in AB, and extend them till the other point reaches 4 in the subdivided inch OA. We have now 2-4 inches in the dividers.

(ii) To get the remaining 7 hundredths, move the right-hand point up the perpendicular through 2 till it reaches the 7th parallel. Then extend the dividers till the left point reaches the diagonal 4 also on the 7th parallel. We have now 2:47 inches in the dividers.

REASON FOR THE ABOVE PROCESS.

The reason of the

The first step needs no explanation. second is found in the Corollary of Theorem 22.

Joining the point 4 to the corresponding point on the tenth parallel, we have a triangle 4,4,5, of which one side 4,4 is divided into ten equal parts by a set of lines parallel to the base 4,5. Therefore the lengths of the parallels between 4,4, and the diagonal 4,5 are 10,10,... of the base, which is I inch.

1

2

3

Hence these lengths are respectively

01, 02, 03, ... of Î inch.

5 4

4 3 2 10

Similarly, by means of this scale, the length of a given straight line may be measured to the nearest hundredth of an inch.

Again, if one inch-division on the scale is taken to represent 10 feet, then 2-47 inches on the scale will represent 24-7 feet. And if one inch-division on the scale represents 100 links, then 2.47 inches will represent 247 links. Thus a diagonal scale is of service in preparing plans of enclosures, buildings, or fieldworks, where it is necessary that every dimension of the actual object must be represented by a line of proportional length on the plan.

NOTE.

The subdivision of a diagonal scale need not be decimal.

For instance we might construct a diagonal scale to read centimetres, millimetres, and quarters of a millimetre; in which case we should take four parallels to the line AB.

[For Exercises on Linear Measurements see the following page.]

EXERCISES ON LINEAR MEASUREMENTS.

1. Draw straight lines whose lengths are 1-25 inches, 2-72 inches, 3.08 inches.

2. Draw a line 2.68 inches long, and measure its length in centimetres and the nearest millimetre.

3. Draw a line 57 cm. in length, and measure it in inches (to the nearest hundredth). Check your result by calculation, given that 1 cm. =0·3937 inch.

4. Find by measurement the equivalent of 3·15 inches in centimetres and millimetres. Hence calculate (correct to two decimal places) the value of 1 cm. in inches.

5. Draw lines 2.9 cm. and 6-2 cm. in length, and measure them in inches. Use each equivalent to find the value of 1 inch in centimetres and millimetres, and ‍take the average of your results.

6. A distance of 100 miles is represented on a map by 1 inch. Draw lines to represent distances of 336 miles and 408 miles.

7. If 1 inch on a map represents 1 kilometre, draw lines to represent 850 metres, 2980 metres, and 1010 metres.

8. A plan is drawn to the scale of 1 inch to 100 links. Measure in centimetres and millimetres a line representing 417 links.

9. Find to the nearest hundredth of an inch the length of a line which will represent 42:500 kilometres in a map drawn to the scale of 1 centimetre to 5 kilometres.

10. The distance from London to Oxford (in a direct line) is 55 miles. If this distance is represented on a map by 2·75 inches, to what scale is the map drawn? That is, how many miles will be represented by 1 inch? How many kilometres by 1 centimetre?

(1 cm. =0 3937 inch; 1 km. = mile, nearly ]

11. On a map of France drawn to the scale 1 inch to 35 miles, the distance from Paris to Calais is represented by 4-2 inches. Find the distance accurately in miles, and approximately in kilometres, and express the scale in metric measure. [1 km. =§ mile, nearly.]

12. The distance from Exeter to Plymouth is 37 miles, and appears on a certain map to be 24"; and the distance from Lincoln to York is 88 km.. and appears on another map to be 7 cm. Compare the scales of these maps in miles to the inch.

13. Draw a diagonal scale, 2 centimetres to represent 1 yard, shewing yards, feet, and inches.