Math 1 · Unit 4 · Lesson 4

Using Slopes to Check Shapes

Put slope to work on the coordinate plane. Given the corners of a shape as points, the slopes of its sides tell you which sides are parallel and which meet at a right angle. That is enough to prove a shape is a rectangle or a right triangle with algebra alone. Equal side lengths need distance, which comes in Unit 11.

About 40 minutes Proof with slope, at the introduction level

The big idea

A shape drawn from its corner points can be tested with the two slope facts from this unit. Compute the slope of each side as rise over run. Sides with equal slopes are parallel, and sides whose slopes multiply to negative one meet at a right angle. So a right triangle shows itself as one pair of perpendicular sides, and a rectangle as two pairs of parallel opposite sides with perpendicular neighbors. That is a proof, because every claim rests on a number you computed rather than on how the picture looks. Slope stops there: it cannot measure a side, so proving a square or comparing lengths waits for distance and midpoint in Unit 11.

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 2Slope can prove a shapeSlope can prove things about shapes. 3Two proofs, workedCorners A(−3, 1), B(1, 3), C(3, −1), D(−1, −3), in order around the shape. 4Plot it, then test itPlot the corners on the grid and check them, then compute the side slopes and decide. 5Practice ProblemsTry them, get help one step at a time, then print how it went.

Printables

One page with the two tests, the rectangle and right-triangle checklists, what slope cannot do, and a worked proof. Plus a worksheet with grids and 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: Same, Parallel, or Perpendicular?The decision this lesson applies to the sides of a shape. Next lesson: From a Line to a RegionUnit 5 begins: the answer is no longer a line but a whole region, with the same boundary line. 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.