Determine, in slope–intercept form,
the equation of the line passing through 𝐴: 13, negative seven perpendicular to the
line passing through 𝐵: eight, negative nine and 𝐶: negative eight, 10.
So, here’s what we’re thinking. We have points 𝐵 and 𝐶, which
form a line. Point 𝐴 does not fall on this
line. But point 𝐴 falls on a line
perpendicular to the line 𝐵𝐶. And we’re trying to find in
slope–intercept form the equation of the line that passes through point 𝐴. Slope–intercept form is the form 𝑦
equals 𝑚𝑥 plus 𝑏. That means we need the slope of
this line and the 𝑦-intercept. But since we don’t know two points
along this line, we’ll have to find the slope a different way.
We remember perpendicular lines
have negative reciprocal slopes. And since we do know two points
along the line 𝐵𝐶, we can find the slope of line 𝐵𝐶. And the slope along the line that
includes point 𝐴 will be equal to negative one over the slope of the line from
𝐵𝐶. This is just a mathematical way to
say that these two values will be negative reciprocals of one another. This means our first job is to find
the slope of line 𝐵𝐶. If we know two points along the
line, we can find their slope by taking 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥
We’ll let 𝐵 be 𝑥 one, 𝑦 one and
𝐶 be 𝑥 two, 𝑦 two. And the slope of line 𝐵𝐶 will be
equal to 10 minus negative nine over negative eight minus eight. 10 minus negative nine is 19. And negative eight minus eight
equals negative 16. We can say that the slope of line
𝐵𝐶 is 19 over negative 16. But more commonly, we would include
the negative sign in the numerator and say the slope of line 𝐵𝐶 equals negative 19
over 16. The slope of the line containing
point 𝐴 is the negative reciprocal of this value.
To find the reciprocal of a
fraction, we flip it. The reciprocal of negative 19 over
16 is 16 over negative 19. But we have to be careful here
because we need the negative reciprocal. And that means negative 16 over
negative 19 simplifies to 16 over 19. The slope of the line passing
through point 𝐴 is then 16 over 19. At this point, we have the slope of
the line passing through point 𝐴. And we have one point that falls
along that line.
To find the 𝑦-intercept form of
this equation, we could then use the point–slope formula, which says 𝑦 minus 𝑦 one
equals 𝑚 times 𝑥 minus 𝑥 one, where 𝑥 one, 𝑦 one is a point along the line. Point 𝐴 is 𝑥 one, 𝑦 one. And so, we have 𝑦 minus negative
seven equals 16 over 19 times 𝑥 minus 13. Minus negative seven is plus
seven. We distribute that 16 over 19 times
𝑥. And 16 over 19 times negative 13
equals negative 208 over 19.
Since we want the equation in
slope–intercept form, we need to get 𝑦 by itself by subtracting seven from both
sides. Negative 208 over 19 minus seven is
negative 341 over 19. This line 𝑦 equals 16 over 19𝑥
minus 341 over 19 is perpendicular to the line 𝐵𝐶 and passes through point 𝐴.