In the figure, the length of 𝐴𝐶 is 3.5. What is the length of 𝐴𝐵? Give your answer to two decimal places.
In this question, we’re given a figure of triangle 𝐴𝐵𝐶, and we’re told that the length of side 𝐴𝐶 is 3.5. We can add this to our diagram. We need to determine the length of the side 𝐴𝐵. We need to give our answer to two decimal places. There’s a few different ways we could go about determining the length of 𝐴𝐵.
To do this, we note that in our diagram we’re given the point 𝐷 which makes two right triangles. We have right triangle 𝐴𝐷𝐵 and right triangle 𝐴𝐷𝐶. And if we add the right angle for triangle 𝐴𝐷𝐶, we note that we have one known side length for this triangle and we know one of the nonright angles of the triangle. This means we can determine the side lengths of this triangle by using right-triangle trigonometry. And in particular, we can use this to find the length of 𝐴𝐷. And then, we’ll be able to use right-triangle trigonometry on our other right triangle 𝐴𝐷𝐶 to determine the length of 𝐴𝐵.
The first thing we’re going to need to do is label the sides of our right triangle 𝐴𝐷𝐶. Let’s start with the hypotenuse. It’s the longest side in the right triangle. It’s the one opposite the right angle. In this case, that’s side 𝐴𝐶. Next, we should label the side we want to calculate; that’s 𝐴𝐷. We can see in our diagram it’s opposite the angle of 41 degrees. So we can label this as the opposite side. Finally, although it’s not necessary, we can also see that side 𝐷𝐶 is adjacent to our angle of 41 degrees. So we can label this as the adjacent side.
Now that we’ve labeled the sides of this right triangle, we’re ready to apply right-triangle trigonometry to determine the length of side 𝐴𝐷. And we can recall the following acronym: SOH CAH TOA. This will help us determine which trigonometric ratio we need to use. In our diagram, we want to determine the opposite side, and we know the length of the hypotenuse. So we need to use the trignometric ratio, which relates the opposite side to the hypotenuse. And this is the sine function. We can recall if 𝜃 is the angle in a right triangle, then the sin of 𝜃 will be equal to the length of the opposite side to angle 𝜃 divided by the length of the hypotenuse. Now, we’d substitute the values from our right triangle 𝐴𝐷𝐶 into this equation. We get sin of 41 degrees is equal to the length of 𝐴𝐷 divided by 3.5.
Now, we can solve for 𝐴𝐷 by multiplying both sides of our equation through by 3.5. We get that the length of side 𝐴𝐷 is 3.5 times the sin of 41 degrees. And now we could use our calculator to evaluate this expression. However, remember, we’re trying to find the length of side 𝐴𝐵. And we’re going to need to use the length of side 𝐴𝐷 to do this, so it’s easier to use the exact value. So we’ll leave this expression as it is.
Let’s now sketch triangle 𝐴𝐵𝐷. We have the triangle 𝐴𝐷𝐵 is a right triangle. We know one of the nonright angles of this triangle and we also know one of the side lengths. So we can determine the length of this triangle by using trigonometry. Once again, we’ll start by labeling the sides of this triangle. We’ll start with the hypotenuse, the longest length of the triangle, the one opposite the right angle. In this case, that’s 𝐴𝐵. Next, we see that the side 𝐴𝐷 is the one opposite our angle of 63 degrees. So we can label this as the opposite side. And finally, although it’s not necessary, we can see that the side 𝐵𝐷 is the one adjacent to the angle of 63 degrees. So we can label this as the adjacent side.
We now need to determine which trigometric ratio we need to use. And to do this, we will once again use our acronym of SOH CAH TOA. We have a very similar story. However, it is slightly different in this case. In triangle 𝐴𝐵𝐷, we know the length of the opposite side to our angle. However, we want to determine the length of the hypotenuse. So we’ll once again be using the sine ratio. Now, all we need to do is substitute the values from triangle 𝐴𝐵𝐷 into the sine ratio. We get sin of 63 degrees will be equal to 3.5 sin of 41 degrees divided by the length of 𝐴𝐵.
Now, all that’s left to do is rearrange this equation for 𝐴𝐵. We’ll multiply both sides of our equation through by 𝐴𝐵 and divide both sides of the equation through by sin of 63 degrees. This gives us that the length of 𝐴𝐵 is equal to 3.5 times the sin of 41 degrees divided by the sin of 63 degrees. And now, we can evaluate this expression by using our calculator, where we need to make sure it’s set to degrees mode. If we do, we get 2.577 and this expansion continues. Well, it’s worth pointing out we could also give this the units of length units since we know this represents the length.
Finally, the question wants us to write our answer to two decimal places. To do this, we look at the third decimal digit of our number, which is seven. Since this is greater than or equal to five, we need to round this value up. And in doing so, we get our final answer. Given in the figure that 𝐴𝐶 is 3.5, we were able to conclude the length of 𝐴𝐵 is 2.58 to two decimal places.