# Video: Finding the Length of a Segment from a Point outside the Circle to the First Intersecting Point on the Circle

In the figure, line segment π΅πΆ is a diameter of the circle π, π΄π΅ = π΄πΆ, πβ π΅π΄πΆ = 60Β°, π΅π = 22.9 cm, and line segment ππ β₯ line segment π΄π΅. Find the length of line segment π΄πΈ.

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### Video Transcript

In the figure below, line segment π΅πΆ is a diameter of the circle π, π΄π΅ equals π΄πΆ, the measure of angle π΅π΄πΆ equals 60 degrees, π΅π equals 22.9 centimeters, and line segment ππ is perpendicular to line segment π΄π΅. Find the length of line segment π΄πΈ.

We want to find how long line segment π΄πΈ is. But first, letβs take the additional information our question gave us and add it into our diagram. We know that π΅πΆ is a diameter of this circle; itβs a chord that passes through the center. This means that line segment ππΆ will be equal in length to line segment ππ΅. We also know that line segment π΄π΅ is equal in length to line segment π΄πΆ. If line segment π΄π΅ is equal to line segment π΄πΆ, this means the angles opposite them will be equal in length. This is a property of triangles.

Since we know that one of the angles is 60 degrees and the other two are equal to each other, this will be an equilateral triangle. After that, we see that π΅π equals 22.9 centimeters. And if we add an additional radius from π to πΈ, we see that that will be equal to the radius from π to π΅. And then, we can say that line segment ππ is a perpendicular bisector of πΈπ΅. And we can say that line segment πΈπ must then also be equal in measure to 22.9 centimeters.

But now, if we consider the fact that this triangle, πΈππ΅, is an isosceles triangle, we can say that the angles opposite the equal sides must be equal to each other, which means those angles must both be 60 degrees. And, of course, if two angles in a triangle are 60 degrees, then the third angle in the triangle must be 60 degrees. How does this help us? Well, it tells us that if we add 22.9 plus 22.9, weβre going to get the radius of the circle. And if from π to π΅ is 45.8 centimeters, then from πΆ to π is 45.8 centimeters. And this means to find the distance from π΄ to πΆ, we just need to add 45.8 plus 45.8. That gives us 91.6 centimeters.

And since π΄π΅ is equal in length to π΄πΆ, π΄π΅ must also measure 91.6 centimeters. But π΄π΅ is made up of two segments, one π΄πΈ and one πΈπ΅. Remember π΄πΈ is our unknown value, but πΈπ΅ will be 22.9 plus 22.9. πΈπ΅ is equal to 45.8 centimeters. And if 91.6 centimeters is equal to π΄πΈ plus 45.8 centimeters, then we subtract 45.8 from both sides. We get that line segment π΄πΈ must be equal to 45.8 centimeters. What this tells us about our circle is that point πΈ divided line segment π΄π΅ in half.