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)²= 3 + 1 - 2√3 = 4 - 2√3

b = √3 + 1

b² = (√3 + 1)²= 3 + 1 + 2√3 = 4 + 2√3

By applying those values in cosine formula

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

(1/2) = (8 - c²)/2(2)

2 = 8 - c²

c²= 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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