Video Transcript
𝐴𝐵𝐶 is a right triangle at 𝐵,
where 𝐵𝐶 equals 10 centimeters and 𝐴𝐶 equals 18 centimeters. Find the length 𝐴𝐵, giving the
answer to the nearest centimeter, and the measures of angles 𝐴 and 𝐶, giving the
answer to the nearest degree.
We haven’t been given a diagram for
this problem. So we need to begin by drawing one
ourselves. We’re told that 𝐴𝐵𝐶 is a right
triangle at 𝐵, which means it’s angle 𝐵 that is the right angle. We’re also told that 𝐵𝐶 is 10
centimeters and 𝐴𝐶 is 18 centimeters. We were asked to find the length
𝐴𝐵 and the measures of each of the other two angles in this triangle.
Let’s begin with finding the length
of 𝐴𝐵. As we have a right triangle in
which we know two of its side lengths, we can apply the Pythagorean theorem to
calculate the length of the third side. The Pythagorean theorem tells us
that, in a right triangle, the sum of the squares of the two shorter sides is equal
to the square of the hypotenuse. In our triangle, the two shorter
sides are 𝐴𝐵 and 𝐵𝐶 and the hypotenuse is 𝐴𝐶. So we have the equation 𝐴𝐵
squared plus 𝐵𝐶 squared is equal to 𝐴𝐶 squared.
Substituting 10 for 𝐵𝐶 and 18 for
𝐴𝐶 gives 𝐴𝐵 squared plus 10 squared is equal to 18 squared. This simplifies to 𝐴𝐵 squared
plus 100 equals 324. So 𝐴𝐵 squared is equal to
224. 𝐴𝐵 is then equal to the square
root of 224, which is 14.9666, or 15 to the nearest integer. So we found the length of 𝐴𝐵. And now we need to consider
calculating the measures of the two angles. Let’s start with angle 𝐴.
We’ll begin by labeling the three
sides of the triangle in relation to this angle. The side directly opposite, that’s
𝐵𝐶, is the opposite. The side between this angle and the
right angle is the adjacent. And the side directly opposite the
right angle is the hypotenuse. We can then recall the acronym
SOHCAHTOA to help us decide which trigonometric ratio to use to calculate this
angle.
As we now know the lengths of all
three sides in the triangle, we could use any of the three ratios. But it makes the most sense to use
the two sides whose lengths we’re originally given in case we made any mistakes when
we calculated the length of 𝐴𝐵. So we’re going to use the sine
ratio. This is defined as sin of 𝜃 is
equal to the opposite over the hypotenuse. Substituting 10 for the opposite
and 18 for the hypotenuse and using 𝐴 to represent the angle at 𝐴, we have sin of
𝐴 is equal to 10 over 18. To calculate 𝐴, we need to apply
the inverse sine function, giving 𝐴 equals the inverse sin of 10 over 18. Evaluating on our calculators,
which must be in degree mode, we have 33.748, which to the nearest degree is 34.
Finally, we need to calculate the
measure of the third angle in the triangle. As the angles in any triangle sum
to 180 degrees, we can do this by subtracting the measures of the other two angles
from 180 degrees, which gives 56 degrees. And so we’ve completed our
solution. The length of 𝐴𝐵 to the nearest
centimeter is 15 centimeters. The measure of angle 𝐴 and the
measure of angle 𝐶, each to the nearest degree, are 34 and 56 degrees,
respectively.
