# Question Video: Finding the Measure of an Angle in a Right-Angled Triangle Given the Lengths of Its Sides Mathematics • 11th Grade

𝐴𝐵𝐶 is a right triangle at 𝐵, where 𝐵𝐶 = 3 cm and 𝐴𝐵 = 4 cm. Find the length of line segment 𝐴𝐶 and the measures of ∠𝐴 and ∠𝐶 to the nearest degree.

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### Video Transcript

𝐴𝐵𝐶 is a right triangle at 𝐵, where 𝐵𝐶 is equal to three centimeters and 𝐴𝐵 is equal to four centimeters. Find the length of line segment 𝐴𝐶 and the measures of angle 𝐴 and angle 𝐶 to the nearest degree.

We will begin by sketching the right triangle 𝐴𝐵𝐶. We are told that side 𝐵𝐶 is three centimeters long and side 𝐴𝐵 is four centimeters long. The first part of our question is to find the length of the line segment 𝐴𝐶. This is the hypotenuse of the triangle, since it is opposite the right angle.

One way of calculating the length of the hypotenuse when given the length of the other two sides of a right triangle is using the Pythagorean theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the hypotenuse and 𝑎 and 𝑏 are the lengths of the two shorter sides. Substituting in the values from this question, we have 𝐴𝐶 squared is equal to three squared plus four squared. Three squared is equal to nine, and four squared, 16. We can then square root both sides of our equation. And since 𝐴𝐶 must be positive, we have 𝐴𝐶 is equal to the square root of nine plus 16. This simplifies to the square root of 25, which is equal to five. The length of the line segment 𝐴𝐶 is five centimeters.

It is worth noting that this triangle is an example of a Pythagorean triple. And as a result, we may simply have recalled that any triangle with side lengths three centimeters, four centimeters, and five centimeters will be a right triangle.

The next part of this question is to find the measures of angles 𝐴 and 𝐶. We will do this using our knowledge of right angle trigonometry and the sine, cosine, and tangent ratios. One way of recalling these ratios is using the acronym SOH CAH TOA, where sin 𝜃 is equal to the opposite over the hypotenuse, cos 𝜃 is equal to the adjacent over the hypotenuse, and tan 𝜃 is equal to the opposite over the adjacent. We will now use one of these ratios to help calculate the measure of angle 𝐴. The side of our triangle that is opposite this angle is 𝐵𝐶, and the side that is adjacent to the angle is the side 𝐴𝐵. We have already labeled the longest side 𝐴𝐶 as the hypotenuse.

As we know all three lengths, we can use any one of the three ratios. In this question, we will choose to use the tangent ratio. The tangent of any angle is equal to the opposite over the adjacent. So, in this question, tan 𝐴 is equal to three over four, or three-quarters. To solve for 𝐴, we take the inverse tangent of both sides such that 𝐴 is equal to the inverse tan of three-quarters. Ensuring that our calculator is in degree mode, we can type this in, giving us an answer of 36.8698 and so on. We are asked to give our answer to the nearest degree, so this is equal to 37 degrees. The measure of angle 𝐴 is 37 degrees.

In order to calculate the measure of angle 𝐶, we could once again use one of our trigonometric ratios. However, it is important to note that side length 𝐴𝐵 is now the opposite, as it is opposite angle 𝐶. In a similar way, side length 𝐵𝐶 is now the adjacent side. Once again, we could use any one of the three ratios. Using the tangent ratio, we have tan 𝐶 is equal to four over three. Taking the inverse tangent of both sides, 𝐶 is equal to the inverse tan of four-thirds. And rounded to one decimal place, this is equal to 53 degrees. The measure of angle 𝐶 is equal to 53 degrees. At this stage, it is worth checking that our three angles sum to 180 degrees, as this is true of any three angles in a triangle.

The three answers to this question are the length of line segment 𝐴𝐶 equals five centimeters, the measure of angle 𝐴 is 37 degrees, and the measure of angle 𝐶 is 53 degrees.