### Video Transcript

π΄π΅πΆ is a right triangle at π΅,
where π΅πΆ equals 10 centimeters and π΄πΆ equals 18 centimeters. Find the length π΄π΅, giving the
answer to the nearest centimeter, and the measure of angles π΄ and πΆ, giving the
answer to the nearest degree.

Letβs begin by sketching this
triangle, which weβre told is a right triangle at π΅. The length of π΅πΆ is 10
centimeters and the length of π΄πΆ is 18 centimeters. We need to find the measures of
both unknown angles and the length of the third side π΄π΅. Letβs begin by calculating the
measure of angle π΄, which we can label as π on our diagram. As we have a right triangle in
which we know two of the side lengths, we can calculate the measure of this angle
using right triangle trigonometry. We begin by labeling the three
sides of the triangle in relation to this angle. π΅πΆ is the opposite, π΄π΅ is the
adjacent, and π΄πΆ is the hypotenuse.

Recalling the acronym SOH CAH TOA,
we can see that it is the sine ratio we need to use because the lengths weβve been
given are the opposite and the hypotenuse. Through an angle π in a right
triangle, the sin of angle π is defined to be equal to the length of the opposite
divided by the length of the hypotenuse. So for this triangle, we have that
sin of π is equal to 10 over 18. To find the value of π, we need to
apply the inverse sine function. So we have that π is equal to the
inverse sin of 10 over 18. Evaluating this on a calculator,
which must be in degree mode, gives 33.7489 continuing. And then rounding to the nearest
degree gives 34 degrees.

So we found the measure of angle
π΄. Now letβs consider how we could
find the measure of angle πΆ. If we wish, we could relabel the
sides of the triangle in relation to this angle. So π΄π΅ becomes the opposite, and
π΅πΆ becomes the adjacent. We could then calculate the measure
of angle πΆ using the cosine ratio. However, itβs more efficient to
recall that angles in any triangle sum to 180 degrees. So to calculate the measure of the
third angle in a triangle, we can subtract the measures of the other two angles from
180 degrees. This tells us that angle πΌ or
angle πΆ to the nearest degree is 56 degrees.

Finally, we need to calculate the
length of side π΄π΅, which we can do using another trigonometric ratio. In relation to angle π or angle π΄
whose measure we know, the side π΄π΅ is the adjacent. Using the cosine ratio, we
therefore have that the cos of 33.7489 continuing degrees is equal to π΄π΅ over
18. Multiplying both sides of this
equation by 18 gives π΄π΅ is equal to 18 cos of 33.7489 degrees. And weβre using the unrounded value
here for accuracy. Evaluating this on a calculator
gives 14.9666 continuing, and rounding this to the nearest integer gives 15. We have then that the length of
π΄π΅ to the nearest centimeter is 15 centimeters. And the measures of angles π΄ and
πΆ, each to the nearest degree, are 34 degrees and 56 degrees.

We can check our answer for the
length of π΄π΅ using the Pythagorean theorem. In a right triangle, the sum of the
squares of the two shorter sides is always equal to the square of the
hypotenuse. If we take the unrounded value for
π΄π΅ and square it and then add 10 squared for π΅πΆ squared, this gives 324. The square of the hypotenuse,
thatβs 18 squared, is also equal to 324. And as these two values are the
same, this confirms that our answer for π΄π΅ is correct. We could also have calculated the
length of π΄π΅ by using the Pythagorean theorem and then checked our answer using
trigonometry.