Video Transcript
In the figure below, line segment
𝐵𝐶 is a diameter of the circle 𝑀, 𝐴𝐵 equals 𝐴𝐶, the measure of angle 𝐵𝐴𝐶
equals 60 degrees, 𝐵𝑋 equals 22.9 centimeters, and line segment 𝑀𝑋 is
perpendicular to line segment 𝐴𝐵. 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 𝐵𝐶 is a diameter of
this circle; it’s a chord that passes through the center. This means that line segment 𝑀𝐶
will be equal in length to line segment 𝑀𝐵. We also know that line segment 𝐴𝐵
is equal in length to line segment 𝐴𝐶. If line segment 𝐴𝐵 is equal to
line segment 𝐴𝐶, 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 𝐵𝑋 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 𝐵. And then, we can say that line
segment 𝑀𝑋 is a perpendicular bisector of 𝐸𝐵. 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, 𝐸𝑀𝐵, 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 𝐵 is 45.8
centimeters, then from 𝐶 to 𝑀 is 45.8 centimeters. And this means to find the distance
from 𝐴 to 𝐶, we just need to add 45.8 plus 45.8. That gives us 91.6 centimeters.
And since 𝐴𝐵 is equal in length
to 𝐴𝐶, 𝐴𝐵 must also measure 91.6 centimeters. But 𝐴𝐵 is made up of two
segments, one 𝐴𝐸 and one 𝐸𝐵. Remember 𝐴𝐸 is our unknown value,
but 𝐸𝐵 will be 22.9 plus 22.9. 𝐸𝐵 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 𝐴𝐵 in half.
