Math 1 · Unit 11 · Lesson 4 · part 2 of 4
Three tools, one checklist
Three tools, three questions: how steep (slope), how long (distance), where is the middle (midpoint).
Three tools, one checklist
Three tools, three questions: how steep (slope), how long (distance), where is the middle (midpoint). Now all three at once. Given a set of points, you can prove what kind of shape they make: show it must be true, using math reasons. Each tool answers one kind of question.
Slope: parallel and perpendicularEqual slopes mean parallel sides. Slopes that are negative reciprocals (they multiply to −1) mean perpendicular sides, a right angle. From Unit 4.
Distance: equal sidesFind the length of each side. Equal lengths mean equal sides.
Midpoint: the diagonalsA diagonal joins two corners that are not next to each other. If both diagonals have the same midpoint, they cut each other in half.
A short checklist.
| shape | what to show | tool |
|---|---|---|
| parallelogram | both pairs of opposite sides parallel, or the diagonals share a midpoint | slope, or midpoint |
| rectangle | a parallelogram with a right angle | slope |
| rhombus | four equal sides | distance |
| square | both a rectangle and a rhombus: four equal sides and four right angles | distance and slope |
| isosceles triangle | two equal sides | distance |
| equilateral triangle | three equal sides | distance |
| right triangle | two perpendicular sides | slope |
Quadrilateral MNPQ: M(3, 0), N(1, 3), P(−2, 1), Q(0, −2). What is it? Use the tools, then decide.
Every side comes out to √13, so the four sides are equal. The slopes alternate −32 and 23, negative reciprocals, so every corner is a right angle. Four equal sides and four right angles: MNPQ is a square.