4. Beginning at B, 100 feet from the bank of a river, a line, BC, is measured 1,200 feet long. At D, distant from C 50 feet, the perpendicular DE is found

to measure 90 feet. distance from B to the opposite bank?

the river?

What is the

A, a tree on

How wide is

5. A boy, whose eye (E) is 4 feet from the ground, can just see the

top (A) of a steeple when he stands back 3 feet from a fence (CG) 6 feet high. The distance from the foot of the fence to the center of the base of the steeple is 177 feet. Find the height of the steeple AB.

6. Wishing to ascertain the distance AB, I measure a line, AD, at right angles to AB, 12 chains;

DE, at right angles to AD, 5 chains; and find that a line sighted from E to B intersects AD at C, distant from D 3.25 chains. What is the distance from A to B?

tower fi, I set up a pole, cd, 12 feet long above the ground. Another pole ab, 44 feet above ground, is set up at such a distance that the tops of the two poles and of the tower are in a line. The distance between the poles (ae or db) is 10 feet. The distance from d to the foot of the tower is 195 feet. The width of the tower (kj) is 30 feet.

α

b d

e

=

kij

The similar triangles aec and ahf give us the proportion ae: ah:: ec : hf. What is the distance ec? ah = bi = bd + dk + ki. ki kj. When fh is found, what must be added to get the height of the tower?

8. To determine the height of a building, MN, a person attached a strip of wood, ab (a tin M

tube or a piece of narrow pipe would be better), to a post, OP, in such a manner that sighting from a, he could just see M, the top of the building. He then sighted down from b, and marked on the ground the point R, on a line with ab.

N

a

R

P

PQ was found by measurement to be 4 feet, RP 6 feet, PN 120 feet. Required MN.

9. Wood-choppers, desiring to know the height of a tree before cutting it, sometimes make an isosceles right-angled triangle of wood or paper, and " step off" the distance on level ground from the point at which they find they can just

see the top of the tree looking along the hypotenuse of the triangle, the c

base being parallel to the ground.

45

How high is the tree AB, if AC is 36 paces of 3 feet each, and the angle ACB is 45°?

10. B is a point on the bank of a stream due east of A on the other bank. A boy walks due south of A until he reaches a point at which he finds, from his pocket compass, that he is directly south-west of B. If the distance AC measures 119 yards, how wide is the stream?

1274. Miscellaneous Exercises.

B

45

45

B

1. Calculate the length in inches of an arc of 60°, the radius of the circle being two inches. Calculate the length of an arc of 120°. Of 180°. Of 240°. Of 300°.

2. Calculate the length in inches of a chord of 60° in the above circle. Of a chord of 120o. Of a chord of 180°. Of a chord of 240°. Of a chord of 300°.

3. In the parallelogram shown in the accompanying diagram, the angle a measures 40°, and the angle b 35°. How many degrees does the angle c contain? angle d? The angle 1? The angle 2?

с

d

a

The

2

4. Inscribe a regular nonagon in a circle of 2 inches radius,

using the protractor.

5. How many degrees does each angle of a regular nonagon contain? Draw a regular nonagon, each side measuring two inches. (Use the protractor.)

6. The distance around a polygon is called its perimeter. What is the perimeter of a regular hexagon, inscribed in a circle whose radius is 1 inch?

What is the circumference of the circle?

7. The distance from the center of a regular polygon to the middle point of one side is called the apothem.

Draw the apothem of a regular hexagon inscribed in a circle of 1 inch radius. About how long is it?

8. Cut a regular hexagon, side one inch, into six triangles. Place three in a line, and fit in the other three so as to make a rectangle. (Divide one of the triangles into two equal parts.)

How long is the base of the rectangle? What part of the perimeter? About how long is the perpendicular of the rectangle?

9. In a circle, radius 1 inch, inscribe a regular octagon. Divide it into eight triangles, and make out of them a rectangle. About what is the half perimeter of the octagon? apothem? Its area?

Its

Which has the greater perimeter, apothem, area, the hexagon or the octagon ?

10. Find the approximate area of a regular hexagon, side 1 inch, apothem about

inch.

11. Find the approximate area of a regular octagon, side about inch, apothem about 1 inch.

12. If we inscribe in a circle a regular polygon of 16 sides, will its perimeter be greater or less than that of the octagon? Which polygon will have the greater apothem?

13. If we inscribe polygons of 32, 64, 128, etc., sides, what will be the greatest perimeter we can have in a circle of 1 inch radius? What will be the greatest apothem?

14. Draw a rectangle that will be about equal to a polygon of a million sides inscribed in a circle whose radius is 1 inch.

Mark upon it the dimensions. Calculate the area.

15. What is the area of a circle whose radius is 2 inches?

16. Find the area of a circle whose diameter is 10 inches.

17. Find the area of a circle whose circumference is 6.2832 inches.

18. Calculate the area of a sector of a circle whose radius is 10 inches, the arc of the sector being 60°.

19. How many square inches are there between the circumferences of two concentric circles whose radii measure 3 and 6 inches, respectively?

SURFACES OF SOLIDS.

1275. Prisms, Cylinders, Pyramids, Cones.

NOTE. The pupils should first examine a number of prisms and pyramids, right and oblique, regular and irregular, triangular, quadrangular, pentagonal, etc. Right and oblique cylinders and cones should also be at hand.

A prism is a body bounded by plane faces, two of which are equal and parallel polygons, the remaining faces being parallelograms.

The two parallel faces of a prism are called its bases. The

remaining faces taken together constitute its convex surface.