Which lines were perpendicular? What about them looks kind of similar?
Lines are perpendicular when they have negative reciprocal slopes.
Practice makes perfect
There were two parts to Exploration 2, let's look at these individually and then compare our results.
Part A
We were given three equations and asked to write each of them in slope-intercept form:
&3x+4y=6 ⇒ y=- 34x+ 32
&3x-4y=12 ⇒ y= 34x-3
&4x-3y=12 ⇒ y= 43x-4
We were then asked to graph them and decide which ones appear to be perpendicular. Let's graph them all on the same coordinate plane.
We can see that the perpendicular lines, the ones most closely resembling a right angle, are:
y=-3/4x+3/2 and y=4/3x-4,
the ones represented by the blue and green lines.
Part B
Once again, we were given three equations and asked to write each of them in slope-intercept form:
&2x+5y=10 ⇒ y=- 25x+2
&-2x+y=3 ⇒ y=2x+3
&2.5x-y=5 ⇒ y=2.5x-5
We can, again, graph the equations and decide which ones appear to be perpendicular. Let's graph them all on the same coordinate plane.
We can see that the perpendicular lines are:
y=-2/5+2 and y=2.5x-5,
the ones represented by the blue and green lines.
Conclusion
In A, the perpendicular lines were:
y=-3/4x+3/2 and y=4/3x-4.
In B, the perpendicular lines were:
y=-2/5+2 and y=2.5x-5.
What do you notice about these pairs? It might not be super obvious, so let's rewrite 2.5 as 52. The slopes of these equations are - 34 with 43 and - 25 with 52. Each time, one is negative, one is positive, and they are just the flipped versions of each other. Those are called negative reciprocals. We can always know if lines are perpendicular because their slopes will be negative reciprocals. This can be modelled by the formula
m_1* m_2=- 1.
