Question

There's one triangle satisfying angle BCA = 60 degrees, AB = 12 and AC = 10 shown below:

The only possible value of BC is a, as shown above. What is a?
Please don't round your answer. Keep it as a square root.

Answer 1

a = 5 + √69

Explanation:

I am just providing another way of solving this. The answer provided by user elzny is absolutely correct and probably easier to understand. He has done a great job.

We can solve for a using the law of cosines which states that given, two sides of a triangle(say a and b) and an included angle (say C), the third side, c can be determined by the relation:

c² = a² + b² -2ab(cosC)

or

a² + b² -2ab(cosC) = c²

If we look at the triangle we see that angle C measures 60° and the two sides that include this angle are of lengths 10 and a. The third side is 12

Plugging in these values into the equation,

10² + a² - 2(10)(a)(cos 60) = 12²

100 + a² -20a(cos 60) = 144

cos 60 = 1/2

So

100 + a² - 20a (1/2) = 144

(100 - 144) + a² - 10a = 0

a² -10a -44 = 0

This is a quadratic equation which can be solved to get two values for a
The solution to this quadratic equation is done using the quadratic formula

`x = ( -b \pm √(b^2 - 4ac))/( 2a )`

where

• a = coefficient of x² = 1 in this problem
• b = coefficient of x = -10 in this problem
• c = constant = -44

Plugging in these values,

`x = ( -(-10) \pm √((-10)^2 - 4(1)(-44)))/( 2(1) )\\\\x = ( 10 \pm √(100 - (-176)))/( 2 )\\\\x = (10)/(2) \pm (√(276))/(2)}`

Factor 276 to get 276 = 4 x 69

`√(276) = √(4 \cdot 69) = √(4) \cdot √(69) = 2 √(69)`

This results in

`x = ( 10 )/( 2 ) \pm (2√(69)\, )/( 2 )\\\\x = 5 \pm √(69)\\\\`

`√(69)` is greater than 5 so we take only the positive component which is

`5 + √(69)`

Which is the same answer as what elzny provided.

The pro of this methodology:
Direct implementation without splitting the triangle

The con:
You have to solve a quadratic equation which is not fun. But most scientific calculators can do this for you

Thanks!

Answer 2

see attached

Explanation:

you need to create 2 right angle triangles

one on the left has 60° which makes it a special right angle triangle with the property of the following:

the side that's opposing the 30° angle will be 1/2 of the long side,

the side that's adjacent to the 30° angle will be √3/2 of the long side

once you know these, the rest is easy