### Video Transcript

π΄π΅πΆ is an isosceles triangle where the side π΄π΅ and the side π΄πΆ are equal to nine centimeters, the line segment π΄π· is perpendicular to the line segment π΅πΆ, and the measure of angle πΆ is 34 degrees. Find the length of the line segment π΅πΆ, giving the answer to one decimal place.

In this question, weβre given some information about an isosceles triangle π΄π΅πΆ. In fact, this is given in a diagram. First, weβre told that sides π΄πΆ and π΄π΅ have equal length. Theyβre both nine centimeters. In the diagram, we can also see that we have a line from π΄ to π·, where π· is a point on the base of the triangle. Thatβs the line segment πΆπ΅. And weβre told that π΄π· is perpendicular to π΅πΆ. So we have two right triangles. π΄π·πΆ and π΄π·π΅ are both right triangles with right angles at π·.

Weβre also told that the measure of angle πΆ is 34 degrees. But remember, this is an isosceles triangle. And since the sides π΄πΆ and π΄π΅ have equal length, the measure of angle πΆ must be equal to the measure of angle π΅. Therefore, the measure of angle π΅ is 34 degrees. We need to use all of this information to determine the length of line segment π΅πΆ. We need to give our answer to one decimal place.

Thereβs many different ways we can go about this. However, the easiest way is to notice that π΄π·π΅ and π΄π·πΆ 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 π΅π· and the length of line segment πΆπ·. Adding these together would then give us the length of the line segment π΅πΆ.

And this would work and give us the correct answer. However, thereβs one extra simplification we can use. We can notice that triangles π΄π·πΆ and π΄π·π΅ 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 π΅π· is equal to the length of πΆπ·. However, we couldβve 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. Weβre going to use triangle π΄π΅π·.

First, weβre 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 π΄π΅; weβll label this as the hypotenuse. Next, we can see that the side π΄π· is opposite the angle 34 degrees, so weβll label this as the opposite side. Finally, we can notice that side π΅π· is adjacent to this angle, so we label side π΅π· as the adjacent side. Weβre 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 π΅πΆ, and weβre doing this by finding the length of the line segment π΅π·, 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 π is an angle in a right triangle, then the cos of π is equal to the length of the side adjacent to angle π 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 π΅π· divided by nine. Finally, we can solve for the length of π΅π· by multiplying through by nine. The length of π΅π· is nine times the cos of 34 degrees. We can evaluate this by using our calculator where we need to make sure itβs set to degrees mode. We get 7.46 and this expansion continues centimeters.

But weβre not done yet. Remember, the question wants us to determine the length of the line segment π΅πΆ. And we can do this by using our diagram. First, the length of π΅πΆ is equal to the length of π΅π· plus the length of π·πΆ. And second, we know that both of these have the same length, so itβs equal to two times the length of π΅π·. And now, weβre going to substitute the exact expression we have for the length of π΅π· into this equation. This is to help prevent rounding errors. We have that π΅πΆ 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 π΅πΆ to one decimal place is 14.9 centimeters.