Why the Sides Match
Part of the Transformations & Constructions section of Coddy's Geometry journey. Lesson 38 of 61.
Why is the constructed triangle equilateral? Because of one rule: Radii are equal. Every radius of a circle, a segment from its center to the circle, has the same length.
In triangle PQR, the sides PQ and PR both run from P to the circle about P, so PR = PQ. The sides QP and QR both run from Q to the circle about Q, so QR = QP. All three sides equal PQ. This is the first construction in Euclid's Elements, written about 2,300 years ago.
The construction is only trusted because each step has a reason, just like an angle chase.
Challenge
C was built from two circles, one about A through B and one about B through A. The sides BC and BA glow. Pick the rule that makes them equal, then press Check.
Try it yourself
Name the rule that links them
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Transformations & Constructions
7Drawing on Squared Paper
Lengths by PythagorasPerpendicular StepsSquares by AreaRight Triangles by AreaMidpoints on the GridRecap: Squared Paper10Perpendiculars and Parallels
Perpendicular From a PointPerpendicular at a PointParallel Through a PointTools You Have EarnedRecap: Parallels5Dilations
Scaling From a CenterShrinking a ShapeDilating About a PointNegative Scale FactorsRecap: Dilations8Compass and Straightedge
Circles Keep DistancesWhy the Sides MatchAngles You BuiltA 60° AngleRecap: Equilateral Triangle