Justification of Geometric Constructions (General)

Role of Justification in Geometric Constructions

Geometric constructions, using only an unmarked straightedge and a compass, are fundamental in Euclidean geometry. While the steps for performing various constructions are often presented as procedures, a critical aspect of formal geometry is the justification of these constructions.

What is Justification?

Justification in geometric constructions is the logical process of demonstrating, using established geometric principles (axioms, postulates, definitions, and previously proven theorems), that the steps performed in a construction procedure unfailingly produce a figure with the desired geometric properties. It answers the question: Why does this construction method work?

Why is Justification Necessary?

1. Mathematical Rigor and Validity: Geometry is built on logical deduction. Every statement must be justified based on accepted truths. Justification ensures that constructions are not merely coincidental drawings that appear correct, but are mathematically valid methods guaranteed to achieve the desired result precisely. It transforms a drawing process into a rigorous mathematical argument.
2. Understanding Underlying Principles: Providing a justification requires analyzing the construction steps in terms of fundamental geometric properties. This process deepens the understanding of concepts like congruence, similarity, properties of lines (parallel, perpendicular), angles, circles, and triangles. For example, justifying the angle bisector construction helps solidify the SSS congruence criterion and the property of points equidistant from angle arms.
3. Distinguishing Construction from Drawing: A drawing is a physical representation and is always subject to inaccuracies of instruments and execution. A geometric construction, when justified, is a theoretical procedure that is perfectly accurate in principle. Justification provides the proof of this theoretical perfection, independent of any specific physical drawing.
4. Ensuring Uniqueness and Correctness: Justification proves that the constructed figure not only possesses the desired properties but is the *unique* figure (up to congruence) defined by the given conditions and the construction steps.
5. Foundation for More Complex Constructions: Basic justified constructions become the building blocks for more complex ones. Knowing *why* a perpendicular bisector works is essential when using it to find a circumcenter or drop a perpendicular.

In essence, justification provides the logical backbone for geometric constructions, ensuring their accuracy, validity, and integration within the broader deductive system of geometry. It shifts the focus from a manual process to a reasoned mathematical outcome.

Using Basic Geometric Principles (Axioms, Postulates, Theorems) for Justification

Justifying a geometric construction involves building a logical argument (a proof) that links the actions taken during the construction to the established rules and facts of geometry. The tools used in these proofs are the foundational elements of Euclidean geometry:

• Axioms or Postulates: These are basic statements accepted as true without proof. In construction, Euclid's postulates regarding the drawing of straight lines and circles are fundamental. For example, the fact that a compass maintains a fixed distance relies on a postulate equivalent to defining a circle.
• Definitions: Precise meanings of geometric terms (e.g., definition of a circle, a midpoint, perpendicular lines, an angle bisector). The justification shows that the constructed figure meets these definitions.
• Theorems: Statements that have been formally proven based on axioms, definitions, and previously proven theorems. Key theorems frequently used in justifications include:
  • Congruence Criteria for triangles (SSS, SAS, ASA, AAS, RHS)
  • Properties of congruent triangles (CPCTC - Corresponding Parts of Congruent Triangles are Congruent)
  • Properties of parallel lines and transversals (corresponding angles, alternate interior angles)
  • Angle properties (Linear Pair Axiom, angles on a straight line, angle sum property of a triangle/quadrilateral)
  • Properties of circles (radius perpendicular to tangent, angle in a semicircle)
  • Properties of specific quadrilaterals (parallelogram diagonals bisect each other, rhombus diagonals are perpendicular bisectors)
  • Basic Proportionality Theorem (Thales' Theorem)

To justify a construction, one typically follows these steps:

1. Clearly state the given information and what needs to be proven (Goal).
2. Describe the construction steps taken.
3. Add necessary auxiliary lines or points to form geometric figures (like triangles or quadrilaterals) that can be analyzed.
4. Use a step-by-step logical argument, citing definitions, axioms, or theorems as reasons, to show that the constructed figure possesses the required properties. Often, proving triangle congruence is a key part of the justification.

Example: Justification for Constructing the Perpendicular Bisector of a Line Segment

Let's use the perpendicular bisector construction (previously described in I3 under "Constructing Perpendiculars") as an example to show how justification is done.

Justification (Proof):

Given: Line segment $AB$. Points $P$ and $Q$ constructed such that $AP = BP = r$ and $AQ = BQ = r$, where $r > \frac{1}{2}AB$. Line $PQ$ intersects $AB$ at $M$.

To Prove: Line $PQ$ is the perpendicular bisector of $AB$. This means we must prove that $AM = MB$ (PQ bisects AB) and $PQ \perp AB$ (PQ is perpendicular to AB, i.e., $\angle PMA = 90^\circ$).

Construction for Proof: Join points $A$ to $P$, $A$ to $Q$, $B$ to $P$, and $B$ to $Q$.

Proof:

Step 1: Prove $\triangle PAQ \cong \triangle PBQ$

Consider $\triangle PAQ$ and $\triangle PBQ$:

Statement	Reason
In $\triangle PAQ$ and $\triangle PBQ$:
$AP = BP$	Arcs of equal radii ($r$) by construction (Step 2 & 3).
$AQ = BQ$	Arcs of equal radii ($r$) by construction (Step 2 & 3).
$PQ = PQ$	Common side to both triangles.
$\triangle PAQ \cong \triangle PBQ$	By SSS (Side-Side-Side) Congruence rule.
$\angle APQ = \angle BPQ$	Corresponding Parts of Congruent Triangles (CPCT). These are the angles $\angle APM$ and $\angle BPM$.

Step 2: Prove $\triangle PAM \cong \triangle PBM$

Now, consider $\triangle PAM$ and $\triangle PBM$. They share the side $PM$, and $A$ and $B$ are on the line $AB$.

Statement	Reason
In $\triangle PAM$ and $\triangle PBM$:
$AP = BP$	By construction (equal radii $r$).
$\angle APM = \angle BPM$	Proved above (CPCT from $\triangle PAQ \cong \triangle PBQ$).
$PM = PM$	Common side to both triangles.
$\triangle PAM \cong \triangle PBM$	By SAS (Side-Angle-Side) Congruence rule.
$AM = BM$	Corresponding Parts of Congruent Triangles (CPCT). This proves that $M$ is the midpoint of $AB$.
$\angle PMA = \angle PMB$	Corresponding Parts of Congruent Triangles (CPCT).

Step 3: Show perpendicularity

Since $AMB$ is a straight line segment (part of line AB), the angles $\angle PMA$ and $\angle PMB$ are adjacent angles that form a linear pair.

$\angle PMA + \angle PMB = 180^\circ$

(Linear Pair Axiom)

From Step 2, we proved that $\angle PMA = \angle PMB$. Substitute $\angle PMA$ for $\angle PMB$ in the above equation:

$\angle PMA + \angle PMA = 180^\circ$

(Substituting $\angle PMB = \angle PMA$)

$2 \angle PMA = 180^\circ$

... (i)

$\angle PMA = \frac{180^\circ}{2}$

... (ii)

$\angle PMA = 90^\circ$

... (iii)

Since $\angle PMA = 90^\circ$, the line $PQ$ is perpendicular to the line segment $AB$.

Conclusion:

We have shown that $AM = BM$ (from Step 2), proving that $PQ$ bisects $AB$.

We have shown that $\angle PMA = 90^\circ$ (from Step 3), proving that $PQ$ is perpendicular to $AB$.

Therefore, line $PQ$ is the perpendicular bisector of line segment $AB$.

Hence Proved (Justified).

This detailed example shows how definitions (midpoint, perpendicular), axioms (Linear Pair), and theorems (SSS and SAS congruence, CPCTC) are woven together to logically demonstrate the correctness of the construction steps.

Verifying the Accuracy of Constructions

While a geometric construction, when performed using ideal tools (perfect straightedge and compass) and justified by proof, is theoretically exact, any physical drawing is subject to imperfections. Verification is the process of checking whether the constructed figure in a physical drawing accurately reflects the properties it is supposed to have. There are two main types of verification:

1. Theoretical Verification (Justification)

As discussed in the previous sections, this is the rigorous mathematical confirmation. It's not about checking a specific drawing, but proving that the construction *method* is correct based on logic and established geometric principles.

• Method: Constructing a formal logical proof.
• Tools: Knowledge of geometric axioms, definitions, and theorems.
• Accuracy: Absolute and mathematically precise. It proves the validity of the construction technique itself.
• Advantage: Provides certainty of the method's correctness.
• Disadvantage: Requires understanding and application of geometric proof techniques.

Justification is the process we have described in detail for various constructions throughout these notes (angle bisector, perpendicular bisector, similar triangles, tangents, etc.).

2. Practical Verification (Measurement)

This involves using measuring instruments (which are marked) to check if the properties of the constructed figure in a physical drawing are approximately correct.

• Method: Using a ruler to measure lengths and a protractor to measure angles.
• Tools: Ruler with markings (e.g., in cm or inches), protractor.
• Accuracy: Approximate. Limited by the precision of the measuring tools, the thickness of the pencil lines, and the human ability to read scales and align instruments accurately.
• Advantage: Quick, intuitive, and provides a tangible check of the drawing. Useful for confirming that no gross errors were made during the construction steps.
• Disadvantage: Cannot provide mathematical proof of correctness. It only verifies the accuracy of the *drawing* in a specific instance, not the theoretical validity of the construction method for all cases.

Examples of Practical Verification:

After performing a construction, you can use a ruler and protractor to check your work:

• Perpendicular Bisector of $AB$ (intersecting at $M$):
  • Measure the lengths $AM$ and $MB$ using a ruler. Are they approximately equal? $AM \approx MB$?
  • Measure the angles $\angle PMA$ or $\angle PMB$ using a protractor. Are they approximately $90^\circ$? $\angle PMA \approx 90^\circ$?
• Angle Bisector of $\angle ABC$ (ray $BD$):
  • Measure the angles $\angle ABD$ and $\angle DBC$ using a protractor. Are they approximately equal? $\angle ABD \approx \angle DBC$?
• Constructing a $60^\circ$ Angle:
  • Measure the constructed angle with a protractor. Is it approximately $60^\circ$?
  • Measure the sides of the constructed equilateral triangle (if applicable). Are they approximately equal?
• Constructing a Triangle given Side Lengths:
  • Measure the lengths of the sides of the resulting triangle with a ruler. Do they match the given lengths?
  • (Optional) Measure the angles with a protractor and check if they sum to $180^\circ$.
• Constructing Tangents from an External Point:
  • After drawing tangent $PT_1$, measure the angle $\angle OT_1P$ using a protractor. Is it approximately $90^\circ$?
  • Measure the lengths of the tangents $PT_1$ and $PT_2$. Are they approximately equal (Property of tangents from an external point)?

Conclusion on Justification vs. Verification

In the context of formal geometry, justification (theoretical verification) is the definitive proof of a construction's correctness. It provides the mathematical certainty that the method is valid for all applicable cases.

Practical verification (measurement) is a pragmatic way to check a specific drawing. It's useful for identifying execution errors but cannot replace the logical validity provided by justification. In educational settings, students are often asked to perform constructions, justify them theoretically, and then verify their specific drawing practically.