Prove $b^2-c^2\leq2a^2$ in $\triangle ABC$

  • Context: MHB 
  • Thread starter Thread starter Albert1
  • Start date Start date
Click For Summary
SUMMARY

The inequality \(b^2 - c^2 \leq 2a^2\) in triangle \(ABC\) with sides defined as \(\overline{AB}=c\), \(\overline{BC}=a\), and \(\overline{CA}=b\) is proven using the Law of Cosines. Given that \(\angle B = 30^\circ\), the Law of Cosines states that \(b^2 = a^2 + c^2 - 2ac \cdot \cos(30^\circ)\). Substituting \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\) into the equation leads to the conclusion that the inequality holds true under the specified conditions.

PREREQUISITES
  • Understanding of triangle properties and definitions
  • Familiarity with the Law of Cosines
  • Knowledge of trigonometric functions, specifically cosine values
  • Basic algebra for manipulating inequalities
NEXT STEPS
  • Study the Law of Cosines in-depth for various triangle configurations
  • Explore trigonometric identities and their applications in triangle inequalities
  • Learn about other triangle inequalities such as the triangle inequality theorem
  • Investigate geometric proofs involving angles and side lengths in triangles
USEFUL FOR

Mathematicians, geometry students, and educators looking to deepen their understanding of triangle inequalities and the application of the Law of Cosines in proofs.

Albert1
Messages
1,221
Reaction score
0
$\triangle ABC$

$(1): \overline{AB}=c,\overline{BC}=a\,\, and \,\,\overline{CA}=b$

$(2)\angle B=30^o$

prove:

$b^2-c^2\leq 2a^2$
 
Mathematics news on Phys.org
Albert said:
$\triangle ABC$

$(1): \overline{AB}=c,\overline{BC}=a\,\, and \,\,\overline{CA}=b$

$(2)\angle B=30^o$

prove:

$b^2-c^2\leq 2a^2$
using law of cosine
$b^2=a^2+c^2-2ac\,cos\,B$
$\therefore b^2-c^2-2a^2=-a^2-\sqrt 3 \,ac\leq 0$
and the proof is done
 

Similar threads

MHB Can AC^2 be Proven to Equal AB(AB+BC) in Triangle ABC with Angles B=80 and C=40?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Using trigonometry to prove <c=50 degree
  • · Replies 1 ·
Replies
1
Views
1K
MHB Proving the Similarity of Two Acute Triangles with Perpendicular Lines
  • · Replies 2 ·
Replies
2
Views
2K
MHB Finding Angle C in Triangle ABC
  • · Replies 3 ·
Replies
3
Views
2K
MHB Solving for $x+y$ in $\triangle ABC$
  • · Replies 3 ·
Replies
3
Views
1K
MHB Finding $\angle ADC$ in $\triangle ABC$
  • · Replies 3 ·
Replies
3
Views
2K
MHB Proving $\angle ABC$ is Acute: Inside the Triangle $ABC$
  • · Replies 4 ·
Replies
4
Views
1K
MHB Can You Solve the Triangle Sides Challenge?
  • · Replies 2 ·
Replies
2
Views
1K
MHB Prove Geometric Sequence with $(a,b,c)$
  • · Replies 1 ·
Replies
1
Views
1K
MHB What is the Value of \(2\tan\frac{1}{2}A\tan\frac{1}{2}B\) in Triangle ABC?
  • · Replies 2 ·
Replies
2
Views
1K