Math 1 · Unit 4 · Lesson 2 · part 2 of 6
Where the rule comes from: a quarter turn
Perpendicular lines cross at a right angle : a square corner, 90 degrees, like the corner of a page.
Where the rule comes from: a quarter turn
Perpendicular lines cross at a right angle: a square corner, 90 degrees, like the corner of a page. Their slopes are related in a way you can build with your own hands, so do not memorize a rule yet. Watch where it comes from.
- 1Start with a lineTake a line that goes up 2 and over 3: slope 23. Draw its staircase: run 3 to the right, rise 2 up.
- 2Turn it a quarter turnRotate the whole staircase 90 degrees to make a square corner with the original. The up and the over trade places: what was a run of 3 now points up, and what was a rise of 2 now points sideways.
- 3Read the new staircaseThe new line rises 3 for a run of 2, and because it now leans the other way, one of those is backwards. Its slope is −32.
The negative reciprocal. Flip the fraction, so the top (numerator) and the bottom (denominator) swap: that is the reciprocal, the multiplicative inverse from Unit 1. Then change the sign. 23 becomes −32. The quarter turn is exactly why: the rise and the run trade places, and the lean reverses.
The quick check23 × (−32) = −1. Two slopes whose product is −1 are always perpendicular. Multiply them; if you get −1, square corner.
More pairsSlope 4 and slope −14 (4 is 41; flip to 14, change the sign). Slope −5 and slope 15. Slope −12 and slope 2. Each pair multiplies to −1.
The special pairA horizontal line (slope 0) and a vertical line (no slope) meet at a right angle too. You cannot flip 0, so handle that pair by sight: flat and straight up and down always make a square corner.