EXAMPLES. XVII. b.

1. At one side of a road is a flagstaff 25 ft. high fixed on the top of a wall 15 ft. high. On the other side of the road at a point on the ground directly opposite the flagstaff and wall subtend equal angles: find the width of the road.

2. A statue a feet high stands on a column 3a feet high. To an observer on a level with the top of the statue, the column and statue subtend equal angles : find the distance of the observer from the top of the statue.

3. A flagstaff a feet high placed on the top of a tower b feet high subtends the same angle as the tower to an observer h feet high standing on the horizontal plane at a distance d feet from the foot of the tower: shew that

(a−b) d2 = (a+b) b2 — 2b2h − ( a − b) h2.

Example II. A flagstaff is fixed on the top of a wall standing upon a horizontal plane. An observer finds that the angles subtended at a point on this plane by the wall and the flagstaff are a and B. He then walks a distance c directly towards the wall and finds that the flagstaff again subtends an angle ß. Find the heights of the wall and flagstaff.

4. A tower standing on a cliff subtends an angle a at each of two stations in the same horizontal line passing through the base of the cliff and at distances of a feet and b feet from the cliff. Prove that the height of the tower is (a+b) tan a feet.

5. A column placed on a pedestal 20 feet high subtends an angle of 45° at a point on the ground, and it also subtends an angle of 45° at a point which is 20 feet nearer the pedestal: find the height of the column.

6. A flagstaff on a tower subtends the same angle at each of two places A and B on the ground. The elevations of the top of the flagstaff as seen from A and B are a and B respectively. If AB=a, shew that the length of the flagstaff is

a sin (a+B-90°) cosec (a-B).

If

Example III. A man walking towards a tower AB on which a flagstaff BC is fixed observes that when he is at a point E, distant c ft. from the tower, the flagstaff subtends its greatest angle. L BEC=a, prove that the heights of the tower and flagstaff are c tan and 2c tan a ft. respectively.

(

π a

2

7. A pillar stands on a pedestal. At a distance of 60 feet from the base of the pedestal the pillar subtends its greatest angle 30°: shew that the length of the pillar is 40/3 feet, and that the pedestal also subtends 30° at the point of observation,

8. A person walking along a canal observes that two objects are in the same line which is inclined at an angle a to the canal. He walks a distance c further and observes that the objects subtend their greatest angle ß: shew that their distance apart is 2c sin a sin ẞ/(cos a+cos B).

9. A tower with a flagstaff stands on a horizontal plane. Shew that the distances from the base at which the flagstaff subtends the same angle and that at which it subtends the greatest possible angle are in geometrical progression.

10. The line joining two stations A and B subtends equal angles at two other stations C and D: prove that

AB sin CBD= CD sin ADB.

11. Two straight lines ABC, DEC meet at C. If
L DAE= L DBE=a, and ▲ EAB=ß, 2 EBC=y,

shew that

BC=

AB sin B sin (a+ß) sin (y-B) sin (a+B+y)*

12. Two objects P and Q subtend an angle of 30° at 4. Lengths of 20 feet and 10 feet are measured from A at right angles to AP and AQ respectively to points R and S at each of which PQ subtends angles of 30°: find the length of PQ.

13. A ship sailing N.E. is in a line with two beacons which are 5 miles apart, and of which one is due N. of the other. In 3 minutes and also in 21 minutes the beacons are found to subtend a right angle at the ship. Prove that the ship is sailing at the rate of 10 miles an hour, and that the beacons subtend their greatest angle at the ship at the end of 3/7 minutes.

14. A man walking along a straight road notes when he is in the line of a long straight fence, and observes that 78 yards from this point the fence subtends an angle of 60°, and that 260 yards further on this angle is increased to 120°. When he has walked 260 yards still further, he finds that the fence again subtends an angle of 60°. If a be the angle which the direction of the fence makes with the road, shew that 13 sin a=5. Also shew that the middle point of the fence is 120 yards distant from the road.

Measurements in more than one plane.

201. In Art. 199 the base line AB was measured directly towards the object. If this is not possible we may proceed as follows.

From A measure a base line AB in any convenient direction in the horizontal plane. At A observe the two angles PAB and PAC; and at B observe the angle PBA.

Let LPAB=a, LPAC=ß,
LPBA=Y,
AB-a, PC=x.

From A PAC,

x= PA sin B.

From Δ ΡΑΒ,

202. To shew how to find the distance between two inaccessible

objects.

Let P and Q be the objects.

Take any two convenient stations A and B in the same horizontal plane, and measure the distance between them.

At A observe the angles PAQ and QAB. Also if AP, AQ, AB are not in the same plane, measure the angle PAB.

At B observe the angles ABP A and ABQ.

In A PAB, we know LPAB, LPBA, and AB;

so that AP may be found.

In A QAB, we know QAB, LQBA, and AB; so that AQ may be found.

In APAQ, we know AP, AQ, and ▲ PAQ;

so that PQ may be found.

H. K. E. T.

13

Example 1. The angular elevation of a tower CD at a place A due South of it is 30°, and at a place B due West of A the elevation

is 18°. If AB=a, shew that the height of the tower is

a

√2+2√5°

From the right-angled triangle DCA, AC=x cot 30°.
From the right-angled triangle DCB, BC=x cot 18°.
But BAC is a right angle,

Example 2. A hill of inclination 1 in 5 faces South. Shew that a road on it which takes a N.E. direction has an inclination 1 in 7.

Let AD running East and West be the ridge of the hill, and let ABFD be a vertical plane through AD. Let C be a point at the foot of the hill, and ABC a section made by a vertical plane running North and South. Draw CG in a N.E. direction in the horizontal plane and let it meet BF in G; draw GH parallel to BA; then if CH is joined it will represent the direction of the road.