Video Transcript
The scale of a map is one
centimeter to 1.35 kilometers. The positions of three towns on a
map form a triangle. Towns 𝐵 and 𝐶 are 17 centimeters
apart, and the angles 𝐶𝐴𝐵 and 𝐴𝐵𝐶 are 83 degrees and 65 degrees,
respectively. Find the actual distance between
towns 𝐴 and 𝐵 and between towns 𝐴 and 𝐶, giving the answer to the nearest
kilometer.
Before we do any tricky
calculations, we can find the measure of angle 𝐶. Since we know that in a triangle
all the angles add up to 180 degrees, the measure of angle 𝐶 equals 180 degrees
minus 83 plus 65 degrees. Therefore, the measure of angle 𝐶
is 32 degrees.
Next, we recognize that we want to
find the actual distances between these towns. And that means we can go ahead and
deal with our scale. On our drawing, the distance
between 𝐵 and 𝐶 is 17 centimeters. Since every centimeter equals 1.35
kilometers, we multiply 17 by 1.35, which means the distance from 𝐵 to 𝐶 is 22.95
kilometers.
It’s helpful to label the triangle
a bit further. The side opposite angle 𝐴 can be
denoted lowercase 𝑎. The side opposite angle 𝐵 can be
denoted lowercase 𝑏. Likewise, the third side is then
lowercase 𝑐. Now, we have a non-right triangle
for which we know all three of its angles and one of its sides. We can use the law of sines to
calculate the lengths of the two missing sides. We know that we can’t use the law
of cosines as that requires two known side lengths.
The law of sines says that 𝑎 over
sin 𝐴 equals 𝑏 over sin 𝐵 which equals 𝑐 over sin 𝐶. Alternatively, that can be written
as sin 𝐴 over 𝑎 equals sin 𝐵 over 𝑏 which equals sin 𝐶 over 𝑐. We could use either form of this
equation. However, since we’re trying to
solve for side lengths, using the first form will result in less rearranging.
We can start by trying to find the
length of side 𝑐. That’s the distance from town 𝐴 to
town 𝐵. Substituting what we know, we get
22.95 over sin of 83 degrees equals 𝑐 over sin of 32 degrees. Make sure to notice that we’re
using the actual distance and not the scale to measure. To solve, we multiply both sides of
the equation by sin of 32 degrees. 𝑐 is therefore equal to 22.95 over
sin of 83 degrees times sin of 32 degrees. Plugging this into the calculator
gives us 12.252 continuing. If you plug this into your
calculator and did not get 12.252 continuing, check that you’re operating in the
degree mode and not into the radian mode.
Correct to the nearest kilometer,
rounding to the ones place, 𝑐 is equal to 12 kilometers. This is the distance from town 𝐴
to town 𝐵.
We’ll follow the same procedure to
find the side length 𝑏, the distance from town 𝐴 to town 𝐶. We’ll use 𝑎 over sin 𝐴 equals 𝑏
over sin 𝐵. Alternatively, we could have used
𝑐 over sin 𝐶 in place of 𝑎 over sin 𝐴. However, we would need to be
careful to make sure we aren’t using rounded numbers that would affect our
accuracy. So we’ll stick with 𝑎 over sin
𝐴.
Plugging in what we know, we get
22.95 over sin of 83 degrees equals 𝑏 over sin of 65 degrees. We’ll solve in the same way by
multiplying both sides of the equation by sin of 65 degrees. We get 𝑏 equals 22.95 over sin of
83 degrees times sin of 65 degrees, which is 20.955 continuing. Rounded to the nearest kilometer,
it’s 21 kilometers. Therefore, the distance from town
𝐴 to 𝐶 equals 21 kilometers.
Using the law of sines, we found
the actual distance between towns 𝐴 and 𝐵 is 12 kilometers to the nearest
kilometer and the actual distance between towns 𝐴 and 𝐶 is 21 kilometers to the
nearest kilometer.
