### Video Transcript

The circle π in the figure has radius seven centimeters. Given that π΄π΅ equals 14.9 centimeters, find the measure of angle π΄ππΆ and ππ΅, rounded to the nearest tenth.

So within this question, weβre asked to find two things, the measure of an angle and the length of a line segment. Letβs begin with the measure of the angle. π΄ππΆ is the angle formed travelling from π΄ to π to πΆ. So itβs this angle that Iβve marked here in orange. This angle is at the center of the circle with two other angles that have been marked, the right angle π΄ππ΅ and the angle of 151 degrees, the angle πΆππ΅. The key fact that we need to remember in order to answer this part of the question is this: the sum of the measures of the central angles in a circle is 360 degrees.

So the sum of the three central angles, angle π΄ππΆ, 90 degrees, and 151 degrees is 360 degrees. We have an equation that we can solve in order to find angle π΄ππΆ. To find angle π΄ππΆ, we need to subtract the other two angles from 360 degrees. Angle π΄ππΆ is equal to 360 degrees minus 90 degrees minus 151 degrees. Itβs equal to 119 degrees. So weβve answered the first part of the question.

Now letβs think about how to answer the second part. Weβre given in the question two pieces of information about the lengths of various lines. π΄π΅ is equal to 14.9 centimeters. And the radius of the circle is seven centimeters. The radius of a circle is a line segment whose endpoints are the center of the circle and a point on the circle itself. In this question, both ππ΄ and ππΆ are radii of the circle. So now that weβve marked all of the lengths onto the diagram, letβs think about how to calculate the length of the line ππ΅.

ππ΅ is a side in the triangle π΄ππ΅ in which we know the lengths of the other two sides. As this triangle is right angled, we can apply the Pythagorean theorem in order to calculate the length of the third side. The Pythagorean theorem tells us that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In this triangle, this means that ππ΅ squared plus seven squared is equal to 14.9 squared. And so we have an equation that we can solve in order to find the length of ππ΅.

Evaluating both seven squared and 14.9 squared gives ππ΅ squared plus 49 is equal to 222.01. Next, we can subtract 49 from both sides of the equation, which gives ππ΅ squared is equal to 173.01. To find ππ΅, we next need to square root both sides of the equation. And using a calculator to do this, gives ππ΅ is equal to 13.15332.

The question has asked for this length to the nearest tenth. And so we need to round our answer. Our answer to the problem then, the measure of angle π΄ππΆ is 119 degrees. And the length of ππ΅ is 13.2 centimeters, to the nearest tenth. Remember, the key fact we used at the start of this question was that the sum of the measures of the central angles in a circle is 360 degrees.