Translate geometric assumptions into algebra while preserving generality and exceptional cases.
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Try
Coordinates turn a geometric claim into equations, but a convenient picture can accidentally restrict the claim. A general triangle can be placed at (0,0),(b,0),(u,v) with b>0 and v≠0. Replacing its third vertex by (0,b) would impose a right isosceles shape.
Predict
Predict which formulas tolerate vertical lines. Midpoints and squared distances work without dividing by a horizontal change. Slope Δy/Δx requires Δx≠0. Vector scalar multiples can express parallel directions even when one or both lines are vertical.
Experiment
Calculate the midpoint of each diagonal in the diagram. For AC, average (0,0) and (11,7); for BD, average (8,0) and (3,7). Both yield (5.5,3.5). Then replace the numerical vertices with parameters to see what caused the agreement.
A parallelogram drawn with equal coordinate scales: A=(0,0), B=(8,0), D=(3,7), C=(11,7). Its diagonals meet at their shared midpoint (5.5,3.5).
Understand
Worked argument: for rectangle A=(0,0), B=(p,0), C=(p,q), D=(0,q), with p,q>0, diagonal AC has squared length p²+q² and diagonal BD has squared length (−p)²+q², the same value. Lengths are nonnegative, so the diagonals are equal. This coordinate placement represents every rectangle up to a rigid motion, which preserves lengths.
Invalid proof: choose p=q=1 and conclude every rectangle has equal diagonals. That checks one square rather than the general rectangle. Another invalid step divides by a slope denominator that might be zero. Keep parameters and handle exceptional orientations explicitly.
Apply
State what your coordinate choice assumes, and why it represents the full class of figures in the theorem. Translate the algebraic result back into geometry: a common midpoint means bisection, and a nonzero scalar multiple means parallel direction. Calculations without that final interpretation leave the proof unfinished.
Continue
Depth: the dot product of two displacement vectors tests perpendicularity without a special vertical-line case. Derive it from squared distances or a right-triangle comparison before using it as a new theorem. Coordinate proof is a method of proof, not a substitute for stating hypotheses. Revisit the distance and midpoint investigation to compare numerical cases with a general coordinate argument.
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Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Find the x coordinate of the midpoint of (−4,7) and (10,−3).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Find the distance from (−2,1) to (4,9).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
What is the slope of the line through (3,−2) and (3,5)?
To prove a theorem for every triangle, which coordinate choice can retain generality?
Let A=(0,0), B=(p,0), D=(q,r), C=(p+q,r), with p>0 and r≠0. Prove the diagonals of this parallelogram bisect each other.
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.
For A=(0,0), B=(b,0), C=(u,v), b>0,v≠0, prove the segment joining the midpoints of AB and AC is parallel to BC and half as long without slopes.
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.
