Ex. 255. From G go 9 miles W. to H, from H go 12 miles N. to A, from A go 17 miles W. to R. What is the distance from G to R?

Ex. 256. A is 12 miles N. of H, D is 24 miles S. of H, O is due W. of A and OH is 42; find OD and OA.

Ex. 257. XT =

19 miles, MX 11 miles, MT = 17.5 miles; how

far is M from the line XT?

HEIGHTS AND DISTANCES.

If a man who is looking at a tower through a telescope holds the telescope horizontally, and then raises (or "elevates") the end of it till he is looking at the top of the tower, the angle he has turned the telescope through is called the angle of elevation of the top of the tower.

If a man standing on the edge of a cliff looks through a horizontal telescope and then lowers (or "depresses ") the end of it till he is looking straight at a boat, the angle he has turned the telescope through is called the angle of depression of the boat.

Remember that the angle of elevation and the angle of depression are always angles at the observer's eye.

If O is an observer and A and P two points (see fig. 14), the angle AOP is said to be the angle subtended at O by AP.

Ex. 258. In fig. 51, name the angles subtended (i) by BD at A, (ii) by AD at B, (iii) by AC at B.

Ex. 259. A vertical flagstaff 50 feet high stands on a horizontal plane. Find the angles of elevation of the top and middle point of the flagstaff from a point on the horizontal plane 15 feet from the foot of the flagstaff.

Ex. 260. The angle of elevation of the top of the spire of Salisbury Cathedral at a point 1410 feet from its base was found to be 16°. What is the height of the spire?

Ex. 261. A torpedo boat passes at a distance of 100 yards from a fort the guns of which are 100 feet above sea-level; to what angle should the guns be depressed so that they may point straight at the torpedo boat?

Ex. 262. From the top of Snowdon the Menai Bridge can be seen, the angle of depression being 4°. The height of Snowdon is 3560 feet. How far away is the Menai Bridge?

Ex. 263. From a point A the top of a church tower is just visible over the roof of a house 50 feet high. If the distance from A to the foot of the tower is known to be 160 yards, and from A to the foot of the house 60 yards, find in feet the height of the tower, and the angle of elevation of its top as seen from A. Ex. 264. A flagstaff stands on the top of a tower. distance of 40 feet from the base of the tower, the angle of elevation of the top of the tower is found to be 231°, and the flagstaff subtends an angle of 25°. Find the length of the flagstaff and the height of the tower.

At a

Ex. 265. At two points on opposite sides of a poplar the angles of elevation of its top are 39° and 48°. If the distance between the points is 150 feet, what is the height of the tree?

Ex. 266. From the top of a mast 80 feet high the angle of depression of a buoy is 24°. From the deck it is 51°. Find the distance of the buoy from the ship.

Ex. 267. At a window 15 feet from the ground a flagstaff subtends an angle of 43°; if the angle of depression of its foot is 11°, find its height.

Ex. 268. A man observes the angle of elevation of the top of a spire to be 23°; he walks 40 yards towards it and then finds the angle to be 29°. What is the height of the spire?

Ex. 269. An observer in a balloon, one mile high, observes the angle of depression of a church to be 35°. vertically for 20 minutes, he observes the angle be now 551°. Find the rate of ascent in miles

G. S.

After ascending of depression to per hour.

Ex. 270. An observer finds that the line joining two forts A and B subtends a right angle at a point C; from C he walks 100 yards towards B and finds that AB now subtends an angle of 107°; find the distance of A from the two points of observation.

Ex. 271. A man on the top of a hill sees a level road in the valley running straight away from him. He notices two consecutive mile-stones on the road, and finds their angles of depression to be 30° and 13° respectively. Find the height of the hill (i) as a decimal of a mile, (ii) in feet.

How To COPY A GIVEN RECTILINEAR FIGURE.

A rectilinear figure is a figure made up of straight lines. An exact copy of a given rectilinear figure may be made in various ways.

1st method. Suppose that it is required to copy a pentagon ABCDE (as in fig. 69). First copy side AB; then ABC; then side BC; then ▲ BCD; etc. You will not find it necessary to

copy all the sides and angles.

Ex. 272. Draw a good-sized quadrilateral; copy it by Method I. If you have tracing paper, make the copy on this; then see if it fits the original. Ex. 273. Repeat Ex. 272, with an (irregular) pentagon.

2nd method. A simpler way is to prick holes through the different vertices of the given figure on to a sheet of paper below; then join the holes on the second sheet by means of straight lines.

3rd method.

Place a sheet of tracing paper over the given figure, and mark on the tracing paper the positions of the different vertices. Then join up with straight lines.

4th method by intersecting arcs.

To copy ABCDE by this method (see fig. 69). Make A'B' = AB. With centre A' and radius equal to AC describe an arc of a circle.

With centre B' and radius equal to BC describe an arc of a circle.

Let these arcs intersect at C'.

Then C' is the copy of C.

Similarly, fix D' by means of the distances A'D' and B'D'; fix E' by means of the distances A'E' and B'E'.

The five vertices A'B'C'D'E' are now fixed, and the copy may

be completed by joining up.

In Ex. 274-276 the copies should be made on tracing paper if possible; the copies can then be fitted on to the originals.

Ex. 274. Draw, and copy (i) a quadrilateral, (ii) a pentagon, by the method of intersecting arcs. If tracing paper is not used, the copy may be checked by comparing its angles with those of the original.

Ex. 275. By intersecting arcs, copy figs. 45 and 48.

Ex. 276. By intersecting arcs, copy the part of fig. 52 which consists of triangles 1, 2, 3, 4, 5, 6.

SYMMETRY.

Ex. 277. Fold a piece of paper once; cut the folded sheet into any pattern you please;

then open it out (see fig. 70).

The figure you obtain is said to be symmetrical about the line of folding. This line is called an axis of symmetry.

. (1)

(2)

fig. 70.

Ex. 278. Make sketches of the symmetrical figures produced when the folded sheet is cut into the following shapes. (Give names if possible.)

d

(i) a rt. ▲ with its shortest side along the crease.
(ii) an isosceles A with its base along the crease.

(iii)

(iv)

(v)

a scalene ▲ with its longest side along the crease.
an obtused with its shortest side along the crease.
a semi-circle with its diameter along the crease.

(vi)

a rectangle with one side along the crease.

(vii) a parallelogram with one side along the crease.

Ex. 279. Which of the following figures possess an axis of symmetry? (You may find that in some cases there is more than one axis.) In each case make a sketch showing the axis (or axes), if there is symmetry. (i) isosceles A, (ii) equilateral, (iii) square, (iv) rectangle, (v) parallelogram, (vi) rhombus, (vii) regular 5-gon, (viii) regular 6-gon, (ix) circle, (x) a semicircle, (xi) a figure consisting of 2 unequal circles, (xii) a figure consisting of 2 equal circles.

s;

Ex. 280. Fold a piece of paper twice (as in Ex. 30), so that the two creases are at right cut the folded sheet into any shape, being careful to cut away all of the original edge of the paper. On opening the paper you will find that you have made a figure with two axes of symmetry at right angles.

Ex. 281.

(1)

B

(2)

fig. 71.

Cut out a paper parallelogram (be careful not to make it a rhombus). Fold it about a diagonal; do the two halves fit?

You will notice that the parallelogram has no axis of symmetry. Yet it certainly has symmetry of some kind. The nature of this symmetry will be made clear by the following exercise.