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Chapter 4: Problem 47

Prove that, in a triangular diagram, where each vertex represents a pure component, the composition of the system at any point inside the triangle is proportional to the length of the respective perpendicular drawn from the point to the side of the triangle opposite the vertex in question. It is not necessary to assume a special case (i.e., a right or equilateral triangle).

Short Answer

Expert verified
Composition is proportional to the length of the perpendiculars to the sides opposite the vertices.

Step by step solution

01

Understanding the Triangle Diagram

The triangular diagram represents a ternary system with three pure components at the vertices, say A, B, and C. Any point inside the triangle represents a mixture with a certain proportion of these components.
02

Identifying Perpendiculars from a Point Inside the Triangle

For any point within this triangle, we can draw a perpendicular line to each side of the triangle. These perpendiculars help in analyzing the composition of the mixture concerning the vertices.
03

Expressing Composition in Terms of Perpendiculars

Let the perpendiculars from point P to the sides of the triangle opposite vertices A, B, and C be denoted as dA, dB, and dC, respectively. The proportions of the components at point P can be linked to these distances.
04

Applying the Area Relationship

The area of a triangle having a vertex as the origin and point P as one of the vertices is proportional to the respective perpendicular distance. The area related to each perpendicular can be compared to the total area of the original triangle to find proportional components.
05

Formulating the Proportional Relationship

Using the area calculations, the component proportions x_A, x_B, and x_C at point P can be expressed as ratios: \ \[ x_A = \frac{dA}{dA + dB + dC}, \ x_B = \frac{dB}{dA + dB + dC}, \ x_C = \frac{dC}{dA + dB + dC} \] illustrating that each component's proportion is proportional to the length of the perpendicular drawn to its opposite side.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangular Diagram
In a ternary system, a triangular diagram serves as a powerful graphical representation. It visually encapsulates the proportions of three components within a mixture. Each of the triangle's vertices denotes a pure component, often labeled as A, B, and C. Any given point inside this triangle reveals a specific composition, with each component's proportion affecting the location of the point. The triangular diagram is thus a vital tool for understanding complex systems, enabling easy visualization and analysis. Broadly, it shows how variations in the composition of the mixture modify its position within the triangle.
Perpendicular Distances
Perpendicular distances play a crucial role in deciphering the triangle diagram's composition. For any point situated inside the triangle, you can draw perpendicular lines from this point to each side of the triangle. These perpendiculars correspond to the sides opposite the vertices A, B, and C.
  • These distances are vital because they provide insight into the relative proportions of each component within the point's mixture.
  • A shorter perpendicular to a side means the mixture is richer in the opposite component.
Consequently, these perpendicular distances are key to understanding how the mixture's composition is balanced and represented within the triangle.
Proportional Composition
Understanding proportional composition involves recognizing how the lengths of these perpendicular distances relate directly to the mixture's component ratios. The point's position inside the triangle is defined by the proportions of components A, B, and C.
  • The formula \( x_A = \frac{dA}{dA + dB + dC} \), \( x_B = \frac{dB}{dA + dB + dC} \), and \( x_C = \frac{dC}{dA + dB + dC} \) expresses this relationship.
  • Each 'x' represents the proportion of a component, corresponding to the length of the perpendicular drawn to the opposing side.
This proportional relationship is central to analyzing and predicting the behavior of the ternary system.
Area Relationship in Triangles
A deep link exists between the areas formed within the triangular diagram and the overall proportions of components in the mixture. Any smaller triangle that involves the original vertex, point P, and any point along the base side can have its area analyzed to determine composition.
  • In detail, the area of such a smaller triangle is influenced by the perpendicular measurement to its base.
  • By comparing these areas, we can similarly deduce the proportions owed to each perpendicular distance.
This geometric approach fortifies the understanding of proportional composition, highlighting why each perpendicular distance holds significant value in describing the triangle's internal point.

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