Question Video: Using Trigonometry to Find the Length of the Base of an Isosceles Triangle Mathematics

Video Transcript

𝐴𝐵𝐶 is an isosceles triangle where the side 𝐴𝐵 and the side 𝐴𝐶 are equal to nine centimeters, the line segment 𝐴𝐷 is perpendicular to the line segment 𝐵𝐶, and the measure of angle 𝐶 is 34 degrees. Find the length of the line segment 𝐵𝐶, giving the answer to one decimal place.

In this question, we’re given some information about an isosceles triangle 𝐴𝐵𝐶. In fact, this is given in a diagram. First, we’re told that sides 𝐴𝐶 and 𝐴𝐵 have equal length. They’re both nine centimeters. In the diagram, we can also see that we have a line from 𝐴 to 𝐷, where 𝐷 is a point on the base of the triangle. That’s the line segment 𝐶𝐵. And we’re told that 𝐴𝐷 is perpendicular to 𝐵𝐶. So we have two right triangles. 𝐴𝐷𝐶 and 𝐴𝐷𝐵 are both right triangles with right angles at 𝐷.

We’re also told that the measure of angle 𝐶 is 34 degrees. But remember, this is an isosceles triangle. And since the sides 𝐴𝐶 and 𝐴𝐵 have equal length, the measure of angle 𝐶 must be equal to the measure of angle 𝐵. Therefore, the measure of angle 𝐵 is 34 degrees. We need to use all of this information to determine the length of line segment 𝐵𝐶. We need to give our answer to one decimal place.

There’s many different ways we can go about this. However, the easiest way is to notice that 𝐴𝐷𝐵 and 𝐴𝐷𝐶 are both right triangles. And we recall if we know one of the non-right angles of a right triangle and one of the side lengths, we can use right triangle trigonometry to determine the other side length in the right triangle. Therefore, by using right triangle trigonometry, we could determine the length of line segment 𝐵𝐷 and the length of line segment 𝐶𝐷. Adding these together would then give us the length of the line segment 𝐵𝐶.

And this would work and give us the correct answer. However, there’s one extra simplification we can use. We can notice that triangles 𝐴𝐷𝐶 and 𝐴𝐷𝐵 are congruent. To do this, we first note that the triangles share two angles and the side length in common. Therefore, by the angle-angle-side congruence criterion, these two triangles are congruent. And in particular, this tells us the length of 𝐵𝐷 is equal to the length of 𝐶𝐷. However, we could’ve also showed this by using trigonometry.

We now want to apply right triangle trigonometry to determine the length of these two sides. To do this, we start by recalling the acronym SOH CAH TOA. This will help us determine which of the three trigonometric ratios we need to use to determine the side length. But first, we need to label the sides of our right triangle based on their position relative to the angle of 34 degrees. We can choose either triangle. We’re going to use triangle 𝐴𝐵𝐷.

First, we’re going to label the hypotenuse as the longest side of the triangle, which is the one opposite the right angle. We can see that this is side 𝐴𝐵; we’ll label this as the hypotenuse. Next, we can see that the side 𝐴𝐷 is opposite the angle 34 degrees, so we’ll label this as the opposite side. Finally, we can notice that side 𝐵𝐷 is adjacent to this angle, so we label side 𝐵𝐷 as the adjacent side. We’re now ready to apply our acronym to determine which of the trigonometric ratios we need to use.

We want to find the length of the line segment 𝐵𝐶, and we’re doing this by finding the length of the line segment 𝐵𝐷, which is our adjacent side. And we only know the length of the hypotenuse in this right triangle. So, we know the length of the hypotenuse and we want to determine the length of the adjacent side. Therefore, we need to use the cosine ratio. We recall if 𝜃 is an angle in a right triangle, then the cos of 𝜃 is equal to the length of the side adjacent to angle 𝜃 divided by the length of the hypotenuse.

Now, we can just substitute these values into this formula. We get the cos of 34 degrees is equal to the length of 𝐵𝐷 divided by nine. Finally, we can solve for the length of 𝐵𝐷 by multiplying through by nine. The length of 𝐵𝐷 is nine times the cos of 34 degrees. We can evaluate this by using our calculator where we need to make sure it’s set to degrees mode. We get 7.46 and this expansion continues centimeters.

But we’re not done yet. Remember, the question wants us to determine the length of the line segment 𝐵𝐶. And we can do this by using our diagram. First, the length of 𝐵𝐶 is equal to the length of 𝐵𝐷 plus the length of 𝐷𝐶. And second, we know that both of these have the same length, so it’s equal to two times the length of 𝐵𝐷. And now, we’re going to substitute the exact expression we have for the length of 𝐵𝐷 into this equation. This is to help prevent rounding errors. We have that 𝐵𝐶 has a length of two times nine cos of 34 degrees. Evaluating this by using our calculator, we get 14.92 and this expansion continues centimeters.

And finally, the question wants us to give our answer to one decimal place. So we look at the second decimal digit, which is two, which is less than five. So we need to round this value down. And this then gives us our final answer. The length of the line segment 𝐵𝐶 to one decimal place is 14.9 centimeters.