Separate a permitted construction procedure from its mathematical correctness.
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Try
Construct an equilateral triangle on a segment AB by drawing one circle centered at A through B and another centered at B through A. Either intersection C has AC=AB and BC=AB because both are radii. The equality is guaranteed by construction, not by measuring the finished triangle.
Predict
Predict what an unmarked straightedge can do. It draws a line through two distinct known points; it does not read lengths or angles. A compass transfers a radius. A construction consists of permitted steps; its proof explains why the resulting object has the desired properties.
Experiment
Use the diagram to plan a perpendicular bisector. Choose the same radius for both endpoint circles, greater than half AB, so there are two distinct intersections P,Q. Join P and Q. If the radius equals half AB there is only one touching point, so the line through two intersections cannot be drawn.
Equal-radius circles centered at A and B intersect at P and Q. Their common chord PQ is the perpendicular bisector of AB. The radius exceeds half AB; this is a construction diagram, not a measurement instruction.
Understand
Worked justification: P and Q are each equidistant from A and B because they lie on equal-radius circles. The locus of points equidistant from A,B is their perpendicular bisector. To justify that locus for a point P off line AB, let M be the midpoint. Congruent triangles AMP and BMP give equal angles at M; these angles are supplementary, so each is 90°. On line AB itself, only M is equidistant. Conversely a point on the perpendicular bisector gives congruent right triangles and equal distances; M is immediate. Thus the line PQ is the required bisector.
Invalid construction: choose different radii at A and B and assume the intersection points are equidistant from the endpoints. Each distance then equals a different radius; the justification fails even if the drawn line appears central.
Apply
For each written construction give the tools, centers, radius choices, intersection choice and joining line. Then prove the result using equal distances or congruence. When two intersections are possible, specify which one serves the intended internal angle or side of a line.
Continue
Depth: for an internal angle bisector, first mark equal distances OA=OB along its rays. Intersect equal-radius circles centered at A,B and choose the intersection opposite O across chord AB. Joining it to O bisects the angle by SSS. Later construction theory asks which exact lengths and angles are constructible; arbitrary angle trisection is not an operation these tools automatically provide.
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Practice and reasoning
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Segment AB is 10 cm long. Equal-radius circles centered at A and B intersect twice only when their radius exceeds what value, in cm?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
An equilateral triangle is constructed on a segment of length 6. What is the square of its altitude?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which action is allowed with an unmarked straightedge?
For the perpendicular-bisector construction with equal-radius circles, why use the line joining their two intersections?
Construct the perpendicular bisector of a segment AB and justify it, including the radius condition.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Construct the internal bisector of an angle strictly between 0° and 180° and prove it using congruent triangles.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
