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 𝐡𝐢 is a diameter of the circle 𝑀, 𝐴𝐡 = 𝐴𝐢, π‘šβˆ π΅π΄πΆ = 60Β°, 𝐡𝑋 = 22.9 cm, and line segment 𝑀𝑋 βŠ₯ line segment 𝐴𝐡. Find the length of line segment 𝐴𝐸.

03:27

Video Transcript

In the figure below, line segment 𝐡𝐢 is a diameter of the circle 𝑀, 𝐴𝐡 equals 𝐴𝐢, the measure of angle 𝐡𝐴𝐢 equals 60 degrees, 𝐡𝑋 equals 22.9 centimeters, and line segment 𝑀𝑋 is perpendicular to line segment 𝐴𝐡. Find the length of line segment 𝐴𝐸.

We want to find how long line segment 𝐴𝐸 is. But first, let’s take the additional information our question gave us and add it into our diagram. We know that 𝐡𝐢 is a diameter of this circle; it’s a chord that passes through the center. This means that line segment 𝑀𝐢 will be equal in length to line segment 𝑀𝐡. We also know that line segment 𝐴𝐡 is equal in length to line segment 𝐴𝐢. If line segment 𝐴𝐡 is equal to line segment 𝐴𝐢, 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 𝐡𝑋 equals 22.9 centimeters. And if we add an additional radius from 𝑀 to 𝐸, we see that that will be equal to the radius from 𝑀 to 𝐡. And then, we can say that line segment 𝑀𝑋 is a perpendicular bisector of 𝐸𝐡. And we can say that line segment 𝐸𝑋 must then also be equal in measure to 22.9 centimeters.

But now, if we consider the fact that this triangle, 𝐸𝑀𝐡, 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, we’re going to get the radius of the circle. And if from 𝑀 to 𝐡 is 45.8 centimeters, then from 𝐢 to 𝑀 is 45.8 centimeters. And this means to find the distance from 𝐴 to 𝐢, we just need to add 45.8 plus 45.8. That gives us 91.6 centimeters.

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

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