𝑋𝑌𝑍 is a triangle, where 𝑌𝑍 is
equal to eight centimeters, the measure of the angle at 𝑌 is 22 degrees, and the
measure of the angle at 𝑍 is 23 degrees. 𝑊 lies on the line 𝑌𝑍, where the
line 𝑋𝑊 is perpendicular to the line 𝑌𝑍. Find the length of the line 𝑋𝑊,
giving your answer to two decimal places.
Let’s start by sketching this
out. Remember, a sketch doesn’t need to
be to scale, but it’s sensible to keep it roughly in proportion so we can check the
suitability of any answers we get. Here we have a non-right-angled,
triangle with the length of one side known. We also know two of the angles.
We can calculate the measure of the
angle at 𝑋 by using the fact that angles in a triangle add to 180 degrees. Subtracting the measure of the
angle at 𝑌 and the measure of the angle at 𝑍 from 180 gives us 180 minus 22 plus
23. The measure of the angle at 𝑋 is
135 degrees. Once we have the measure of the
angle at 𝑋, we can use the law of sins to calculate either of the missing
sides. We know to use this over the law of
cosines since that requires at least two known sides.
The law of sins says that 𝑎 over
sin 𝐴 equals 𝑏 over sin 𝐵, which equals 𝑐 over sin 𝐶. That’s sometimes written as sin 𝐴
over 𝑎 equal sin 𝐵 over 𝑏, which equals sin 𝐶 over 𝑐. We can use either of these
forms. However, since we’re trying to find
the missing side, it’s sensible to use the first form to minimize the amount of
rearranging we need to do. We would use the second form if
we’re trying to calculate the measure of one of the angles.
We can change the letters in our
formula to suit the problem. In that case, it’s 𝑥 over sin 𝑋,
𝑦 over sin 𝑌, and 𝑧 over sin 𝑍. We can also label the sides of the
triangle as shown. The side opposite angle 𝑋 is
lowercase 𝑥, the side opposite angle 𝑌 is lowercase 𝑦, and the side opposite 𝑍
is lowercase 𝑧.
At this stage of the problem, we
can choose whether we want to calculate the length of the side marked 𝑦 or the side
marked 𝑧. Let’s choose the side marked
𝑧. We’re going to use the formula 𝑥
over sin 𝑋 equals 𝑧 over sin 𝑍. Substituting all the values we know
into the formula gives us eight over sin 135 equals 𝑧 over sin 23.
To solve this equation, we’ll
multiply both sides by sin of 23. 𝑧 is equal to eight over sin of
135 multiplied by sin of 23, which is 4.42 centimeters. We won’t round this answer just
yet. Instead, we’ll use its exact form
in our next calculations.
We were told that the line 𝑋𝑊 is
perpendicular to the line 𝑌𝑍, so we now have a right-angled triangle 𝑋𝑊𝑌. We can use right angle trigonometry
to find the length of the line 𝑋𝑌. Let’s call that 𝑎. Labeling our triangle, we can see
that the side 𝑋𝑊 is opposite to the angle. And we’ve calculated the length of
the side 𝑋𝑌, which is now the hypotenuse of our triangle. This means we need to use the sine
ratio to calculate the length that we’ve called 𝑎.
Substituting the values we know
into this formula gives us sin of 22 is equal to 𝑎 over 4.42. And to solve this equation to
calculate the value of 𝑎, we’ll multiply both sides by 4.42. We’ll try and use the exact value
we calculated earlier. 𝑎 is equal to sin of 22 multiplied
by 4.42, which is 1.6559 and so on. Correct to two decimal places, the
length of 𝑋𝑊 is 1.66 centimeters.