Solving Triangle ABC: Finding Value of x

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Homework Help Overview

The discussion revolves around solving for the variable x in triangle ABC, where the sides are defined in terms of x and one angle is given. The triangle has sides AB = 2x cm, AC = x cm, BC = 21 cm, and angle BAC = 120 degrees. Participants are exploring the application of the cosine rule to find the value of x.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • One participant attempts to apply the cosine rule using angle B, while another questions whether using angle A would be more straightforward. There is a discussion about the correct application of the cosine rule and the implications of using the known angle.

Discussion Status

Some participants have provided guidance on applying the cosine rule correctly, emphasizing the importance of using the known angle A. There is an ongoing exploration of the calculations involved, with one participant claiming to have found a value for x, but seeking confirmation of its correctness.

Contextual Notes

There is a mention of potential confusion regarding the labeling of angles and sides, as well as the need for clarity in the application of the cosine rule. The original poster's calculations are noted, but there is no consensus on the correctness of the final value presented.

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Homework Statement


In triabgle ABC, AB = 2x cm, AC = x cm, BC = 21 cm and angle BAC = 120 degrees. Calculate the value of x?

Homework Equations


I used the cosine rule here

The Attempt at a Solution


As we have 2 sides and an angle I used the cosine rule - Am I correct?
(2) means squared

b(2) = a(2) + c(2) - 2ac cos B
so
x(2) =21(2) + (2x)(2) - 2.21.2x.c cos B
x(2) =441 + 4x(2) - 84x cos B
3x(2) = 84x cosB - 441

How do I find x from here?
 
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Why did you choose to use angle B ?
Since you know angle A, wouldn't that be easier?
 
Natasha1 said:

Homework Statement


In triabgle ABC, AB = 2x cm, AC = x cm, BC = 21 cm and angle BAC = 120 degrees. Calculate the value of x?

Homework Equations


I used the cosine rule here

The Attempt at a Solution


As we have 2 sides and an angle I used the cosine rule - Am I correct?
(2) means squared

b(2) = a(2) + c(2) - 2ac cos B
so
x(2) =21(2) + (2x)(2) - 2.21.2x.c cos B
x(2) =441 + 4x(2) - 84x cos B
3x(2) = 84x cosB - 441

How do I find x from here?
Applying the cosine rule is the way to go.
But you should apply it correctly, with the angle you know (that's angle A):
##a²=b²+c²-2bc\cos(A)##, where a is the side opposite angle A, b the side opposite angle B and c the side opposite angle C.
You also know ##\cos(A)=\cos(120°)##.
 
Last edited:
Thanks everyone, I have found that x(2) = 63 which makes x = 7.94 cm (to 2dp) using the cosine rule

Am I correct?

a(2) = b(2) + c(2) - 2bc cos B
so
21(2) =x(2) + (2x)(2) - 2.x.2x cos B
441 = 5x(2) - 4x(2) cos 120
441 = x(2) (5-4 cos 120)
x(2) = 441 / (5-4cos 120)
x(2) = 63
x = 7.94 cm (to 2dp)
 
Natasha1 said:
Thanks everyone, I have found that x(2) = 63 which makes x = 7.94 cm (to 2dp) using the cosine rule

Am I correct?

a(2) = b(2) + c(2) - 2bc cos B
so
21(2) =x(2) + (2x)(2) - 2.x.2x cos B
441 = 5x(2) - 4x(2) cos 120
441 = x(2) (5-4 cos 120)
x(2) = 441 / (5-4cos 120)
x(2) = 63
x = 7.94 cm (to 2dp)
Looks correct (except that the angle should better be called A, not B).
 

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