In triangle \(ABC\) we have \((a+b+c)(a+b-c)=3ab,\) where \(a,b,\) and \(c\) are the sides opposite \(A, B,\) and \(C\) respectively. What is the degree measure of \(\angle C\)?
Am confused by question
What do you mean by sides OPPOSITE of A, B, C? is there a picture?
In triangle ABC we have (a+b+c)(a+b-c)=3ab, where a, b and c are
the sides opposite A, B and C respectively.
What is the degree measure of \(\angle\ C=\angle\ \gamma\) ?
\(\color{BrickRed}(a+b+c)(a+b-c)=3ab\\ a²+ab-ac+ab+b²-bc+ac+bc-c²-3ab=0\\ a²-ab+b²=c² \)
\(c²=a²+b²-\color{blue}ab\\ c²=a²+b²-\color{blue}2ab \cdot cos (\gamma)\ \color{red}(was\ wrong\ before:\ -2bc\cdot cos(\gamma))\\ 2ab\cdot cos(\gamma)=ab\\ cos(\gamma)= \frac{ab}{2ab}\\ cos(\gamma)=\frac{1}{2}\)
was corrected by me
\(\gamma=arc\ cos\ ( \frac{1}{2})\)
\(C=\gamma=60°\)
be inapplicable:
[If it is possible to determine the ratio ( \(\large \frac{a}{c}\) \( |\ c\ge 2a\) ) from equation\( (a+b+c) (a+b-c) = 3ab\), \(\angle\ \gamma\) can also be determined with the equation \(\gamma=arc\ cos\ ( \frac{a}{2c})\).]
Danke CPhill!
!
( a + b + c) (a + b - c) = 3ab
[ (a + b) + c ] [(a + b) - c ] = 3ab
(a + b)^2 - c^2 = 3ab
a^2 + 2ab + b^2 - c^2 = 3ab
a^2 + b^2 = ab + c^2 (1)
Using the Law of Cosines
c^2 = [a^2 + b^2] - 2(a)(b) cos C (2)
Sub (1) into (2) and we have that
c^2 = [ ab + c^2] - (2ab) cos C
c^2 = ab + c^2 - (2ab) cos C subtract c^2 from both sides
0 = ab - 2ab cos C subtract ab from both sides
-ab = - 2ab cos C divide both sides by -2ab
1/2 = cos C
arccos (1/2) = C = 60°
Thanks to asinus for the hint of applying the Law of Cosines !!!