Video Transcript
In the diagram, 𝐴𝐵 equals six and
𝐵𝐷 equals five. Find 𝐴𝐶 and find 𝐶𝐷.
A good place to start is listing
out the information we’re given. We have triangle 𝐴𝐵𝐶. And in that triangle, line segment
𝐵𝐷 is equal to line segment 𝐷𝐶. We’ve also been told that the
measure of angle 𝐴𝐷𝐶 equals 90 degrees. It’s a right angle. We also know that 𝐴𝐵 measures six
and 𝐵𝐷 measures five. From this information, we can draw
some conclusions. First of all, we already know that
line segment 𝐷𝐶 is equal in length to line segment 𝐵𝐷, which means we can say
that line segment 𝐷𝐶 also measures five. And we can write the line segment
either way. 𝐶𝐷 will be equal to 𝐷𝐶.
So we can say, first of all, that
𝐶𝐷 equals five. To find 𝐴𝐶, we’ll need to think a
little bit more carefully about what we know. We know that the measure of angle
𝐴𝐷𝐶 is 90 degrees, and we know that the point 𝐷 is halfway between 𝐵 and
𝐶. This means we can say that 𝐴𝐷 is
a perpendicular bisector of line segment 𝐵𝐶. We can say that this is true based
on the definition of a perpendicular bisector. A perpendicular bisector has to
divide the line segment in half and meet at a 90-degree angle.
And since we know that line 𝐴𝐷 is
a perpendicular bisector of 𝐵𝐶, we can apply the perpendicular bisector theorem,
which tells us that if a point is on the perpendicular bisector of a line segment,
then it is equidistant from the endpoints. Since 𝐴𝐷 is a perpendicular
bisector, 𝐴𝐵 is equal in length to 𝐴𝐶. And if line segment 𝐴𝐶 is equal
in length to line segment 𝐴𝐵, since line segment 𝐴𝐵 measures six, line segment
𝐴𝐶 will also measure six.
