3. In Fig. 5, A B = 18.7m, BC and BD is the bisector of the

DC. (Log.)

29.4m, A C = 40.4",

A B C ; find A D and

4. If, in Fig. 5, A D = 3 feet 5 inches, A B = 4 feet 2 inches, and B C = 7 feet, find the length of A C.

5. In Fig. 6, CD is the bisector of the ACF, BE 3.3m, A C = 6d, BC= 4.1dm; find A B in yards. (Log.)

6. If, in Fig. 6, A C = 35 yards; find B E in

7. If, in Fig. 6, A E

=

65 yards, A B 48 yards, B C metres. (Log.)

18 feet 6 inches, B C = 14 feet, and B E 14 feet 2 inches; find in metres the lengths of A C and A B.

=

(Log.)

8. The sides of a ▲ are a = 15", b= 12", c = 10m; find the segments into which each side is divided by the bisector of the opposite .

9. Find the segments into which each side is divided by the bisector of an exterior in the preceding problem.

10. The homologous sides of two similar A are 5 feet 3 inches and 4 feet 5 inches, respectively. If the altitude to the given side of the first is 3 feet 9 inches, find the homologous altitude in the second.

11. The sides of a are 4m 6dm, 6m 1dın, and 8m; the homologous sides of a similar are a, 305cm, c; find a and c.

12. In the ▲ A B C and A' B' C', A = 59° = A', b = 3 feet 6 inches, c = 13 feet, b′ = 5.6", c′ = 20.8m. Show what relation, if any, these A bear to each other.

13. The perimeters of two similar polygons are 88m and 396, respectively. One side of the first is 15 yards 4 feet 2.4 inches; find the homologous side of the second. (Log.)

14. The sides of two A are, respectively, 4Km, 9Km, 11Km, and 1.2 miles, 2.7 miles, 3.3 miles. Show by your work any relation which may exist between these A.

=

15. One of the altitudes of a ▲ 1.5m; find the homologous altitude of a similar A, if the perimeters of the two are respectively 15 feet and 24 feet.

16. A series of straight lines passing through the point O intercept segments, on one of two parallel lines, of 15 feet, 18 feet, 24 feet, and 32 feet, the segment of the other parallel, corresponding to 24 feet, is 16 feet; find the other segments.

17. Two homologous sides of two similar polygons are 35 and 50m, respectively. The perimeter of the second is 8Hm. What is the perimeter of the first?

18. The legs of a right ▲ are 3m and 4m; find, in inches, the difference between the hypotenuse and the greater leg. Find also the segments of the hypotenuse made by the perpendicular from the vertex of the right ; and this perpendicular itself.

19. In a whose diameter is 16m, find the length of the chord which is 4m from the centre.

20. The sides of a are 30cm, 40cm, and 45cm; find the projection of the shortest side upon the longest.

21. Is the of 20 acute, right, or obtuse? Which would it be if the sides were 30cm, 40cm, 55cm ? Find the

projection of the shortest side upon the medium side in the latter A.

22. A tangent to a whose radius is 1 foot 6 inches, from a given point without the circumference, is 2 feet; find the distance from the point to the centre.

the

23. In the▲ A B C, a = 14TM, b =

Cacute, right, or obtuse?

= 17m, c = 22m; is

24. To find the altitude of a ▲ in terms of its sides.

A 4

B

α

(1) h2 = c2 – B D3. (The square of either leg of a right A is equal to the square of the hypotenuse minus the square of the other leg.)

(b+a−c) (b−a+c) ̧

2a

2a

2(s—c) × 2(s—a) __ 4s(s—a) (s—b) (s—c).

2

a

2

a2

Extracting the square root, h= √ s(s-a) (s—b) (s·−c),

Similarly,

and

h'=z √s(s—a) (s—b) (s—c),

h' and h" representing the altitude of the

c, respectively.

upon b and

25. To find the radius of the circumscribed in terms of the sides of the A.

C

FIG. 8.

FIG. 9.

୪

ac=2Rx B D. (Fig. 8.)

(The product of two sides of a A is equal to the diameter of the circumscribed multiplied by the altitude to the third side.)

2

But by 24, B D= √ s(s—a) (s—b) (s—c)'

26. To find the bisectors of the of a in terms of the sides.

(1) a c=x2+A DxD C. (Fig. 9.)

(The product of two sides of a is equal to the square of the bisector of the included, plus the product of the segments of the third side made by the bisector.)

Transposing in (1), (2) a2=a c-A DxD C.

(The bisector of an of a divides the opposite side into segments proportional to the adjacent sides.)