Math 1 · Unit 11 · Lesson 4

Proving Shapes with Coordinates

Three tools, three questions, all at once. Slope shows parallel sides (equal slopes) and right angles (negative reciprocals). Distance shows equal sides. Midpoint shows whether the diagonals cut each other in half. With a short checklist, a set of points can be proved to be a parallelogram, rectangle, rhombus, or square, or an isosceles, equilateral, or right triangle. A proof is a few clear sentences that name the tool and the result.

About 45 minutes Name the tool, state the result

The big idea

Three tools answer three questions, and together they prove what a set of points makes. Slope tells parallel, when two sides have equal slopes, and perpendicular, when the slopes are negative reciprocals that multiply to negative 1. Distance tells equal sides. Midpoint tells whether the diagonals cut each other in half. A parallelogram has both pairs of opposite sides parallel, or diagonals that share a midpoint; a rectangle is a parallelogram with a right angle; a rhombus has four equal sides; a square is both. Triangles are isosceles with two equal sides, equilateral with three, and right with two perpendicular sides. Shared midpoints also find a missing corner, the missing endpoint idea from Lesson 2. And a proof is a few clear sentences that name the tool and state the result, each one something a reader can check.

The parts of this lesson

Each part is its own short page. Go in order, or pick the one you need.

1The Words We UseThe vocabulary for this lesson, in one place.Start here 2Three tools, one checklistThree tools, three questions: how steep (slope), how long (distance), where is the middle (midpoint). 3The missing corner, and how to write a proofRectangle ABCD has A(1, 2), B(6, 0), and C(10, 10). 4Practice ProblemsTry them, get help one step at a time, then print how it went.

Printables

One page with the three tools and the checklist, MNPQ as a square, the missing corner by shared midpoints, and how to write a proof sentence. Plus a worksheet with an answer key and a short lesson plan.

Study guide (PDF) Worksheet and answer key (PDF) Lesson plan (PDF)

Where this goes next

Previous lesson: Perimeter and Area from CoordinatesDistance for every side. Next lesson: Pictures of Data and Their ShapesUnit 12 begins: one quantity at a time, and three questions every time. Back to the Math 1 mapEvery Math 1 lesson, in course order. Tell us what you thoughtA few quick questions about this lesson. It takes a minute, and it is how the lesson gets better.

NC Math 1 standards in this lesson: NC.M1.G-GPE.4 (verify algebraically that a set of points makes a particular type of triangle or quadrilateral), NC.M1.G-GPE.5 (slope criteria for parallel and perpendicular lines), NC.M1.G-GPE.6 (midpoints and endpoints).