Video Transcript
𝐴𝐵𝐶 is a right triangle at
𝐵. The point 𝐷 lies on the ray 𝐵𝐶,
where 𝐶𝐷 equals 17 centimeters, the measure of angle 𝐴𝐷𝐶 equals 46 degrees, and
the measure of angle 𝐶𝐴𝐷 equals 24 degrees. Find the length of segment 𝐴𝐵,
giving your answer to the nearest centimeter.
The clue here is that point 𝐷 lies
on the ray 𝐵𝐶. There’s actually no way that the
point can lie between 𝐵 and 𝐶 and still be 46 degrees. Let’s sketch this out and see what
it looks like.
Notice that we know two of the
angles in the triangle 𝐴𝐶𝐷 and the length of one of its sides. This means we can use the law of
sines to calculate the length of side 𝐴𝐶. That’s the side shared by both
triangles. We know that we’ll use the law of
sines over the law of cosines, since the law of cosines requires at least two known
side lengths.
Let’s begin by labeling the sides
of the triangle. We’ll label the side opposite angle
𝐴 with the lowercase 𝑎 and the side opposite angle 𝐶 as lowercase 𝑐, finally,
the side opposite angle 𝐷 as lowercase 𝑑. We can rewrite the equation as 𝑎
over sin 𝐴 equals 𝑑 over sin 𝐷 since we know 𝑎 over sin 𝐴 and we’re trying to
find the side length 𝑑. Plugging in the relevant
information gives us 17 over sin of 24 degrees equals 𝑑 over sin of 46 degrees. Multiplying through by sin of 46
degrees gives us 𝑑 equal to 17 over sin of 24 degrees times sin of 46 degrees. Plugging this into our calculator
gives us 30.065 continuing centimeters. And since we’ll be using this value
again, we won’t round it just yet. We want to round to the nearest
centimeter in our final step.
We’ve calculated the length of
sin [side] 𝐴𝐶. That’s the hypotenuse of the right
triangle 𝐴𝐵𝐶. We’re looking for the length of
line segment 𝐴𝐵. To do that, we’ll need to consider
the angle 𝐵𝐶𝐴. We know that all the angles in a
triangle must add up to 180 degrees. And therefore, the measure of angle
𝐴𝐶𝐷 is 110 degrees. And since the angle 𝐵𝐶𝐴 and the
angle 𝐴𝐶𝐷 together equal 180 degrees, 𝐵𝐶𝐴 equals 70 degrees. It’s also true that the acute angle
𝐶 will be equal to the sum of angles at 𝐴 and 𝐷. The exterior angle in a triangle
will always be equal to the sum of the two interior opposite angles. And that’s 70 degrees.
Back to our triangle 𝐴𝐵𝐶, the
length 𝐴𝐶 is the hypotenuse of this right angle triangle. And since we’re trying to find the
length of 𝐴𝐵 using the angle at 𝐶, 𝑥 is the opposite side length. Based on that, we’ll use the sine
ratio to solve. The sin of 𝜃 equals the opposite
over the hypotenuse. Therefore, the sin of 70 degrees
equals 𝑥 over 30.065. Multiplying through by 30.065
nonrounded, 𝑥 will be equal to sin of 70 degrees times 30.065 continuing. Plugging that into our calculator
gives us 28.252 continuing. This is a measure of
centimeters. That value to the nearest
centimeter is 28. The length of segment 𝐴𝐵 is 28
centimeters.
