# DETERMINE WHETHER THE FOLLOWING MEASUREMENTS PRODUCE A TRIANGLE

## About "Determine Whether the Following Measurements Produce a Triangle"

Determine Whether the Following Measurements Produce a Triangle :

Here we are going to see some example problems to determine whether the following measurements produce a triangle.

## Determine Whether the Following Measurements Produce a Triangle - Examples

Question 1 :

Determine whether the following measurements produce one triangle, two triangles or no triangle:

∠B = 88°, a = 23, b = 2. Solve if solution exists.

Solution :

Let us apply the given details in sine formula

a/sin A  =  b/sin B  =  c/sin C

23/sin A = 2/sin 88°  =  c/sin C

23/sin A  =  2/sin 88°

sin A  =  (23/2) sin 88°

sin A  =  11.5 sin 88°

The maximum value of sin θ will be 1. So, it is not possible.

Hence the given measurements will not produce a triangle.

Question 2 :

If the sides of a triangle ABC are a = 4, b = 6 and c = 8, then show that 4 cosB + 3cosC = 2.

Solution :

Let us use cosine formula to find the values of cos B and cos C.

Cosine formula :

cos A  =  (b2 + c2 - a2)/2bc  --------(1)

cos B  =  (a2 + c2 - b2)/2ac  --------(2)

cos C  =  (a2 + b2 - c2)/2ab  --------(3)

From (2)

cos B  =  (a2 + c2 - b2)/2ac

cos B  =  (42 + 82 - 62)/2(4)(8)

cos B  =  (16 + 64 - 36)/64

cos B  =  44/64  =  11/16

From (3)

cos C  =  (a2 + b2 - c2)/2ab

cos C  =  (42 + 62 - 82)/2(4)(6)

=  (16 + 36 - 64)/48

=  -12/48

cos C  = - 1/4

Given that :

4 cos B + 3cos C = 2

L.H.S :

=  4(11/16) + 3(-1/4)

=  (11/4) - (3/4)

=  (11 - 3)/4  =  8/4

=  2  --> R.H.S

Hence it is proved.

Question 3 :

In a triangle ABC, if a = 3 − 1, b = 3 + 1 and C = 60°, find the other side and other two angles

Solution :

cos C  =   (a2 + b2 - c2)/2ab

a = 3 − 1, b = 3 + 1

a2  =  (3 − 1)= 3 + 1 - 23  =  4 - 23

b = 3 + 1

b2  =  (√3 + 1)2  = 3 + 1 + 2√3  =  4 + 2√3

By applying those values in cosine formula

cos 60  =  (4 - 23 + 4 + 23 - c2)/2(3 − 1)(3 + 1)

(1/2)  =  (8 - c2)/2(2)

2  =  8 - c2

c=  8 - 2  =  6

c = √6

Sine formula :

a/sin A  =  b/sin B  =  c/sin C

(3 − 1)/sin A  =  (3 + 1)/sin B  =  √6/sin 60°

(3 + 1)/sin B  =  √6/(√3/2)

(3 + 1)/sin B  =  2 √6/√3

(3 + 1)/sin B  =  6√2/3

(3 + 1)/sin B  =  2√2

sin B  =  (3 + 1)/2√2

sin B  =  (3/2) (1/√2) + (1/2)(1/√2)

=  sin 60 cos 45 + cos 60 sin 45

=  sin (60 + 45)

=  sin 105

Hence angle B is 105.

A + B + C  =  180

A + 105 + 60  =  180

A + 165  =  180

A  =  180-165

A  =  15

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