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How do you use coordinates to prove a figure is a particular shape?

Use coordinates to prove simple geometric facts about triangles and quadrilaterals using slope and distance (NC.M1.G-GPE.4).

An NC Math 1 EOC answer on coordinate proofs (NC.M1.G-GPE.4): using slope to show sides are parallel or perpendicular and the distance formula to show sides are congruent, to classify triangles and quadrilaterals.

Generated by Claude Opus 4.811 min answer

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Jump to a section
  1. What this topic is asking
  2. The two tools
  3. Proving a triangle is right
  4. Classifying a quadrilateral
  5. How the NC Math 1 EOC examines this topic
  6. Why coordinates turn geometry into algebra
  7. Try this

What this topic is asking

NC.M1.G-GPE.4 asks you to use coordinates to prove simple geometric facts about triangles and quadrilaterals: for example, showing a triangle is right (perpendicular sides), isosceles (two congruent sides), or that a quadrilateral is a parallelogram or rectangle. The tools are slope (for parallel and perpendicular) and the distance formula (for congruent sides).

The two tools

Coordinate proofs rest on slope and distance.

Proving a triangle is right

Classifying a quadrilateral

For four vertices, test opposite sides:

  • Both pairs of opposite sides parallel (equal slopes): parallelogram.
  • Parallelogram with a right angle (a perpendicular pair of adjacent sides): rectangle.
  • All four sides congruent (equal distances): rhombus.

You usually compute slopes (for parallel/perpendicular) and, if needed, distances (for congruence) and read off the classification.

How the NC Math 1 EOC examines this topic

  • Multiple choice. Choose what a figure is, given vertices or computed slopes and lengths.
  • Short reasoning. State why a triangle is right or a quadrilateral is a parallelogram.
  • Technology-enhanced. Match figures to properties, or select all true statements.

These proofs combine slope criteria and the distance formula, the two core coordinate tools.

Why coordinates turn geometry into algebra

Placing a figure on the coordinate plane converts geometric questions into calculations you can check. "Is this angle a right angle?" becomes "do these two slopes multiply to −1-1?"; "are these sides equal?" becomes "are these two distances equal?". This is powerful because algebra is exact and leaves no room for an eyeballed guess: a figure that merely looks like a rectangle is only proven to be one when the slopes show perpendicular sides and the distances confirm the shape. G-GPE.4 is the first taste of this coordinate method, which later courses extend, and the discipline it teaches, pick the tool that matches the claim, then compute, is exactly what the EOC rewards.

Try this

Q1. Sides ABAB and CDCD both have slope 34\frac{3}{4}. Are they parallel? [1 point]

  • Cue. Equal slopes, so yes, parallel.

Q2. A triangle has two sides of length 20\sqrt{20} and one of length 66. What kind is it? [2 points]

  • Cue. Two equal sides: isosceles.

Exam-style practice questions

Practice questions written in the style of NCDPI exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

NC Math 1 EOC (style)2 marksA triangle has vertices A(0,0)A(0, 0), B(4,0)B(4, 0), C(0,3)C(0, 3). Show it is a right triangle.
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It is a right triangle because sides ABAB and ACAC are perpendicular.

Find the slopes meeting at AA. Side ABAB from (0,0)(0,0) to (4,0)(4,0) is horizontal (slope 00). Side ACAC from (0,0)(0,0) to (0,3)(0,3) is vertical (undefined slope). A horizontal and a vertical line are perpendicular, so the angle at AA is a right angle, making it a right triangle. Using slopes to prove a right angle is the G-GPE.4 method.

NC Math 1 EOC (style)2 marksA quadrilateral has vertices forming opposite sides with slopes 2,22, 2 and −13,−13-\tfrac{1}{3}, -\tfrac{1}{3}. What can you conclude?
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Both pairs of opposite sides are parallel, so it is a parallelogram.

Opposite sides with equal slopes are parallel. One pair has slope 22 and the other has slope −13-\frac{1}{3}, and within each pair the slopes match, so both pairs of opposite sides are parallel. A quadrilateral with both pairs of opposite sides parallel is a parallelogram. (The slopes are not negative reciprocals, so it is not a rectangle.)

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