Lesson 3.4: Applications of Geometric Proofs

From Theorems to Real Problems

Apply congruent triangles and rigid motions to measure inaccessible distances and construct congruent shapes using classical tools.

Learning Objectives

Model real problems with congruent triangles
Construct congruent triangles with compass & straightedge
Explain solution validity via proofs
Communicate reasoning clearly

Worked Example (River Width)

Select point A across the river, pick B on near bank, draw BC $\perp$ AB, mark midpoint O of BC, draw CD $\perp$ BC such that A, O, D are collinear. Show $CD=AB$ .

$\angle ABO=\angle DCO=90^\circ$ , $BO=CO$ , $\angle AOB=\angle DOC$ (vertical). Therefore △ABO ≅ △DCO by ASA, hence $AB=CD$ .

More Real-World Scenarios

A) Surveying: Distance to an Island

Set a baseline on shore, create congruent triangles via reflections/rotations to transfer distances; justify with congruence.

B) Construction: Copy a Triangle

Given △ABC and a ray from P, construct △PQR ≅ △ABC using compass-straightedge.

1) Draw PQ on ray with PQ = AB.

2) With centers P and Q, radii AC and BC, intersect to get R.

3) Prove △PQR ≅ △ABC by SSS.

C) Mapping: Rigid-Motion Alignment

Align a satellite image feature to a map by a composition of translation, rotation, and reflection; argue invariants preserved.

Common Pitfalls

Skipping justification of why two constructed segments are equal.
Using similarity when the task requires congruence (ensure scale preserved).
Assuming perpendicularity from diagrams; must be proved or constructed.

Practice

1) Construct a triangle congruent to △ABC given side lengths and included angle.

Hint/Solution

Recreate included angle at P, lay off adjacent sides with compass, intersect arcs → SAS.

2) Use shadow lengths and similar triangles to estimate a building height; justify steps.

Hint/Solution

Establish angle equality (parallel sun rays), set proportion, compute height; discuss measurement error.

3) Given a river and two accessible points on one bank, design a construction to measure the distance to a point across the river using congruent triangles.

Hint

Emulate the "river width" technique with perpendiculars and midpoints to transfer length.

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