### Video Transcript

The line segment π΄π΅ is a chord in
circle π whose radius is 25.5 centimeters. If the length of π΄π΅ is 40.8
centimeters, what is the length of the line segment π·πΈ?

The line segment π·πΈ is the
segment highlighted in orange. Itβs the line that joins the point
π·, which is inside the circle, to the point πΈ on the circle circumference. Notice that this is a segment of
the line ππΈ, which connects the center of the circle to the point on the
circumference, and is therefore a radius of the circle. The length of the line segment ππΈ
then is 25.5 centimeters as this is the radius of the circle given in the
question.

We could therefore work out the
length of the line segment π·πΈ by subtracting the length of ππ· from ππΈ. So we need to consider how we could
work out the length of ππ·. Letβs join in a line connecting the
points π and π΅ together. π is the center of the circle. And π΅ is a point on the
circumference. Itβs the endpoint of our chord
π΄π΅. And therefore, ππ΅ is the radius
of the circle, which means its length is 25.5 centimeters as given in the
question.

We now have a right-angle triangle,
triangle ππ·π΅. And we know the length of the
hypotenuse. Weβre also told in the question
that the length of the chord π΄π΅ is 40.8 centimeters. And we can use this information to
work out the length of the line segment π΅π·.

We know that the perpendicular
bisector of a chord passes through the center of the circle. Now, the line ππ· or ππΈ does
pass through the center of the circle. It passes through the point π. And from the diagram, we know that
it meets the chord π΄π΅ at right angles. Therefore, by the converse of this
statement, it must be the perpendicular bisector of the chord π΄π΅. So this tells us that the lengths
of π·π΅ and π·π΄ are equal because the chord has been bisected. We can therefore find each of these
lengths by halving the length of the chord π΄π΅. 40.8 divided by two is 20.4.

We now know two of the lengths in
our right-angle triangle. So we can apply the Pythagorean
theorem in order to work out the third length ππ·, which is one of the two shorter
sides. 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 our triangle, that means that
ππ· squared plus 20.4 squared is equal to 25.5 squared. And we can solve this equation to
find the length of ππ·. 20.4 squared is 416.16. And 25.5 squared is 650.25. Subtracting 416.16 from each side
of this equation gives that ππ· squared is equal to 234.09.

To find the value of ππ· then, we
need to take the square root of each side of this equation, taking only the positive
square root as ππ· is a length. We have that ππ· is equal to the
square root of 234.09, which is 15.3.

So now that weβve calculated the
length of ππ·, we can return to our calculation for the length of π·πΈ, which we
said we were going to work out by subtracting the length of ππ· from ππΈ, which
was the radius of the circle. Substituting the lengths for ππΈ,
25.5, and ππ·, 15.3, we have that the length of π·πΈ is equal to 25.5 minus 15.3,
which gives 10.2. The units for this are centimeters
because the radius of the circle was also given in centimeters.

So by recalling that the
perpendicular bisector of a chord passes through the center of the circle and then
applying the Pythagorean theorem, we found that the length of the line segment π·πΈ
is 10.2 centimeters.