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Geometry
8.7 Relation Between the Sides of a Triangle and T-Ratios

The Law of sines
In any triangle, the sides are proportional to the sines of the opposite angles, i.e.,
a/sin A = b/sin B = c.sin C
Proof: Let ABC be any oblique triangle. Draw the altitude CD = h. Two cases are possible; all angles are acute or one angle obtuse.
In either case;
Sin B = h/a from the first figure
h = a sin B
From the second figure
sin A = h/b
h = b sin A

Equating the two values of h we get, asin B = bsin A => a/sin A = b/sin B
Similarly by drawing perpendicular from A to BC, we can prove that
c/sin C = b/sin B
Therefore in any triangle a/sin A = b/sin B = c/sin C

Relations between three sides and two angles
In any triangle, to prove that
a = b cos C + c cos B, b = c cos A + a cos C, c = a cos B + b cos A
In the first figure we have
AD/AC = cos A , hence AD = AC cos A = b cos A
DB/BC = cos B, hence DB = BC cos B = a cos B
Now, AB = AD + DB = b cos A + a cos B => c = a cos B + b cos A
Similarly, we can prove the other relations.

The law of cosines
The square on any side of a triangle is equal to the sum of the squares on the other two sides minus twice the product of those two sides and the cosine of the included angle, i.e,
a2 = b2 + c2 - 2bc cos A
b2 = c2 + a2 - 2ca cos B
c2 = a2 + b2 - 2ab cos C
Proof: Consider any oblique triangle ABC with the altitude CD drawn from the vertex C to the opposite side.

From the first figure we have
AD = b cos A, CD = b sin A, DB = AB - AD = c-b cos A
From figure 2 we have
cos A = cos (180 - ang CAD) = -cos CAD = - AD/b => AD = -b cos A; CD = b sin A;
DB = DA + AB = c - b cos A
Thus in both figures, DB = c - b cos A, CD = b sin A
Hence a2 = DB2 + CD2
(Pythagoras theorem)
= (c-b cos A)2 + b2 sin 2A
=> a2 = c2 - 2bc cos A + b2(cos2A + b2Sin2A).
Hence a2 = b2 + c2 - 2bc cos A
Other results can be proved similarly.
Cos A = b2 + c2 - a2/2bc
Cos B = c2 + a2 - b2/2ca
Cos C = a2 + b2 - c2/2ab

Area of a triangle
A = s(s-a)(s-b)(s-c)1/2
s = (a+b+c)/2
where a, b and c are sides of the triangle .


Directions: Solve the following.
Q 1: In triangle ABC, a = 8.4 cm, b = 7.6 cm and c = 9.2 cm. Calculate the area of triangle ABC.
45 sq cm
50 sq. cm
30 sq cm

Q 2: In triangle ABC, a = 50, c = 60, b = 100 degrees, find b.
84.5
85
80

Q 3: In triangle ABC, if b = 6, c = 4, cos A = 1/12, find a
31/2
4.31/2
4

Q 4: In triangle ABC , if 2 cos A/a + cos B/b + 2 cos C/c = a/bc + b/ca. find angle A
30 degrees
45 degrees
90 degrees

Q 5: Solve the triangle ABC given that A = 67o, b = 3 cms, c = 2 cms.
a = 5 cm, B = 70o19l, C = 39o 41l
a = 2 cm, B = 73oC = 39o 41l
a = 2.88 cm, B = 73o19l, C = 39o 41l

Q 6: In a triangle ABC, a = 2, b = 3 and sin A = 2/3, find angle B.
90 degrees
75 degrees
80 degrees

Q 7: The sines of the angles of a triangle are in the ratio of 4:5:6, find the ratio of the cosines of the angles.
12:9:2
2:3:4
4:5:2

Q 8: The angles of a triangle ABC are in AP and b:c = 31/2:21/2. Find angle A.
45 degrees
75 degrees
60 degrees

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Question 10: This question is available to subscribers only!


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