17. The sides of a right-angled triangle are 21 and 28 feet; find the length of the perpendicular drawn to the hypothenuse from the right angle.

18. If in any triangle the angles be to one another as 12:3, prove that the corresponding sides are as 1 : 3:2.

C
2

find tan and prove that in this triangle a+c=2b.

20. In an isosceles right-angled triangle a straight line is drawn from the middle point of one of the equal sides to the opposite angle. that it divides the angle into parts whose cotangents are 2 and 3.

Shew

21. The perpendicular AD to the base of a triangle ABC divides it into segments such that BD, CD and AD are in the ratio of 2, 3 and 6; prove that the vertical angle of the triangle is 45°.

22. A ring, ten inches in diameter, is suspended from a point one foot above its centre by 6 equal strings attached to its circumference at equal intervals. Find the cosine of the angle between consecutive

strings.

23. If a2, b2 and c2 be in A.P., prove that cot A, cot B and cot C are in A. P. also.

B

24. If a, b and c be in A. P., prove that cos A cot cos B cot C

2

2

and cos C cot

are in A. P.

26. The sides of a triangle are in A.P. and the greatest and least angles are @ and ø; prove that

27. The sides of a triangle are in A.P. and the greatest angle exceeds the least by 90°; prove that the sides are proportional to 7+1, √√7 and √7 − 1.

29. In any triangle ABC if D be any point of the base BC, such that BD: DC :: m : n, prove that

and

(m+n) cot ADC=n cot B-m cot C,

(m+n)2 AD2 = (m+n) (mb2+nc2) — mna2.

30. If in a triangle the bisector of the side c be perpendicular to the side b, prove that

2 tan A+ tan C=0.

31. In any triangle prove that, if e be any angle, then

b cos 0=c cos (A − 0) + a cos (C+0).

32. If p and q be the perpendiculars from the angular points A and B on any line passing through the vertex C of the triangle ABC, then prove that

a2p2+b2q2-2abpq cos C=a2b2 sin2 C.

33. In the triangle ABC, lines OA, OB, and OC are drawn so that the angles OAB, OBC, and OCA are each equal to w; prove that

and

cotw=cot A + cot B+ cot C,

cosec2 w=cosec2 A + cosec2 B+ cosec2 C.

CHAPTER XIII.

SOLUTION OF TRIANGLES.

174. IN any triangle the 3 sides and the 3 angles are often called the elements of the triangle. When any 3 elements of the triangle are given, provided they be not the 3 angles, the triangle is in general completely known, i.e. its other angles and sides can be calculated. When the 3 angles are given, only the ratios of the lengths of the sides can be found, so that the triangle is given in shape only and not in size. When 3 elements of a triangle are given the process of calculating its other 3 elements is called the Solution of the Triangle.

We shall first discuss the solution of right-angled triangles, i.e. triangles which have one angle given equal to a right angle.

The next four articles refer to such triangles, and C denotes the right angle.

175. Case I. Given the hypothenuse and one side, to solve the triangle.

Let b be the given side and c the given hypothenuse.
The angle B is given by the

A

The angle A (= 90° – B) is then known.

The side a is obtained from either of the relations

b α

a

cos B

tan B or a = √(c − b) (c+b).

C

176. Case II. Given the two sides a and b, to solve the triangle.

Here B is given by

Hence L tan B, and therefore B, is known.

The angle A (= 90° — B) is then known.

The hypothenuse c is given by the relation c=Va2 +b2. This relation is not however very suitable for logarithmic calculation, and c is best given by

177. Case III. Given an angle B and one of the sides

178. Case IV. Given an angle B and the hypothenuse

1. In a right-angled triangle ABC, where C is the right angle, if a=50 and B=75°, find the sides. (tan 75°=2+ √3.)

2. Solve the triangle of which two sides are equal to 10 and 50 feet and of which the included angle is 90°; given that log 20=1.30103, and L tan 26° 33′ = 9.6986847, diff. for 1'=3160.

3. The length of the perpendicular from one angle of a triangle upon the base is 3 inches and the lengths of the sides containing this angle are 4 and 5 inches. Find the angles, having given

log 2=30103, log 3=4771213,

L sin 36° 52′ = 9.7781186, diff. for 1'=1684,

L sin 48° 35′-9·8750142, diff. for 1'1115.

4. Find the acute angles of a right-angled triangle whose hypothenuse is four times as long as the perpendicular drawn to it from the opposite angle.