From Altitudes to Angles to Sides

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SUMMARY

The discussion focuses on solving a triangle's dimensions using Heron's formula, given the relationships between the area and the triangle's sides. The area is expressed as $A=21a=24b=56c$, allowing for the derivation of sides $a$, $b$, and $c$ in terms of the area $A$. By substituting these expressions into Heron's formula, the area can be calculated, leading to the determination of the triangle's sides as $112/3 \sqrt{3}$, $98/3 \sqrt{3}$, and $14 \sqrt{3}$.

PREREQUISITES
  • Understanding of Heron's formula for triangle area calculation
  • Basic algebra for manipulating equations
  • Knowledge of triangle properties and relationships between sides and area
  • Familiarity with square roots and their simplification
NEXT STEPS
  • Study the derivation and application of Heron's formula in various triangle scenarios
  • Explore advanced geometric properties of triangles, including altitudes and their relationships to sides
  • Learn about the implications of triangle similarity and congruence on side lengths
  • Investigate other area calculation methods for triangles, such as the formula involving base and height
USEFUL FOR

Mathematicians, geometry students, and educators looking to deepen their understanding of triangle properties and area calculations using Heron's formula.

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I don't know where to startView attachment 6411
 

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If you are allowed to use the Heron's formula, then you can do the following. Let $AB=c$, $BC=a$ and $AC=b$. Then the area of the triangle is $A=21a=24b=56c$. Express $a$, $b$ and $c$ though $A$ and substitute in the Heron's formula. You will get an equation in $A$, from where $A$ can be found. Then it is easy to find sides from altitudes.
 
Is it 112/3* \sqrt{3}, 98/3* \sqrt{3}and 14* \sqrt{3}
 

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