Video Transcript
Let 𝑀 be the line on points zero,
negative eight and negative four, 10 and 𝐿 the perpendicular to 𝑀 that passes
through the origin zero, zero. What is the measure of the positive
angle that 𝐿 makes with the positive 𝑥-axis? Give your answer to the nearest
second.
In this question, we are given some
information about two lines 𝑀 and 𝐿. We are given two points that 𝑀
passes through. And we are told that 𝐿 is the
perpendicular to 𝑀 that passes through the origin. We want to use this information to
determine the measure of the positive angle that 𝐿 makes with the positive
𝑥-axis. We need to give our answer to the
nearest second.
To answer this question, we can
begin by recalling a relationship between the measure of an angle that a line makes
with the positive 𝑥-axis and its slope. We can recall that if 𝛼 is the
measure of an angle that the line makes with the positive 𝑥-axis, then the tan of
𝛼 must be equal to the slope the of the line. This also holds true for vertical
lines if we allow the case where 𝑀 is undefined. We can use this relationship to
find the value of 𝛼. It is worth noting that there are
infinitely many angles that a line makes with the positive 𝑥-axis, and they are all
solutions to this equation. We only want the angle with the
smallest positive measure.
Therefore, we want to find the
slope of line 𝐿. We can do this by using the fact
that it is perpendicular to line 𝑀. We can find the slope of line 𝑀 by
using the slope formula: 𝑦 sub two minus 𝑦 sub one over 𝑥 sub two minus 𝑥 sub
one, where these are the coordinates of two distinct points on line 𝑀. We can substitute the given
coordinates of the two points on line 𝑀 into the formula to obtain 𝑀 equals 10
minus negative eight over negative four minus zero. We can then evaluate this
expression to see that the slope of line 𝑀 is negative nine over two. We can find the slope of line 𝐿 by
noting it is perpendicular to line 𝑀.
We then recall that we find the
slope of a perpendicular line by taking negative the reciprocal of the slope. Therefore, the slope of 𝐿 is two
over nine. Another way of thinking about this
is to say that the product of the slopes of lines 𝑀 and 𝐿 must be negative
one.
We can now substitute the slope of
this line into the formula. We do need to be careful to
substitute the correct slope of line 𝐿 into this formula, since this is line we
want to analyze. Substituting this into the formula
gives us that the tan of 𝛼 must be equal to two over nine. Since this is positive, we can
solve for 𝛼 by taking the inverse tangent of both sides of the equation. Therefore, 𝛼 is equal to the
inverse tan of two over nine, which we can calculate is 12.52 and this expansion
continues degrees. At this point, it is always worth
double-checking that 𝛼 is positive as required.
We can convert this value into
degrees, minutes, and seconds by pressing the conversion button on our
calculator. We obtain 12 degrees, 31 minutes,
and 43.71 seconds to the nearest hundredth of a second. Rounding this to the nearest second
gives us that the line 𝐿 makes an angle of 12 degrees, 31 minutes, and 44 seconds
with the positive 𝑥-axis to the nearest second.
