𝐴𝐵𝐶 is a triangle where the
measure of the angle at 𝐴 is 30 degrees and the measure of the angle at 𝐵 is 105
degrees. Find the ratio of lengths 𝑎 to 𝑏
Let’s start by sketching this
diagram out. Remember, the diagram doesn’t need
to be to scale, but it should be roughly in proportion so we can check the
suitability of any answers we get. The measure of the angle at 𝐴 is
30 degrees, and the measure of the angle at 𝐵 is 105 degrees. Since angles in a triangle add to
180 degrees, we can calculate the measure of the angle at 𝐶 by subtracting 105 and
30 from 180 degrees. 180 minus 105 plus 30 is 45, so the
measure of the angle at 𝐶 is 45 degrees.
Now we know the size of all the
angles in our triangle, we should label its sides. The side opposite the angle 𝐴 is
denoted by lowercase 𝑎. The side opposite angle 𝐵 is
lowercase 𝑏. And the side opposite angle 𝐶 is
given by lowercase 𝑐.
We have a non-right-angled triangle
with three known angles. We can use the law of sines to help
us answer this question. Let’s substitute what we know into
the formula for the law of sines. That gives us 𝑎 over sin 30 is
equal to 𝑏 over sin 105, which is equal to 𝑐 over sin 45. Sin 30 is a half, sin 105 is root
six plus root two over four, and sin 45 is root two over two. So we can rewrite our equation as
shown. We’ve shown that the lengths of the
side in the triangle are proportional to the sines of their opposite sides, so we
can rewrite this equation as a ratio.
Next, we need to simplify this
ratio as far as possible. We can do this by multiplying every
part of the ratio by four. The ratio simplifies to two to root
six plus root two to two root two.