### Video Transcript

Given that ๐ธ๐ถ equals four, ๐ธ๐ท
equals 15, and ๐ธ๐ต equals six, find the length of the line segment ๐ธ๐ด.

Letโs look at the diagram more
closely. We can see that it consists of a
circle, in which there are two intersecting chords. Theyโre the lines ๐ด๐ต and
๐ถ๐ท. Weโve also been given various
lengths that we can add to our diagram. The length of the line segment ๐ธ๐ถ
is four. The length of the line segment ๐ธ๐ท
is 15. And the length of the line segment
๐ธ๐ต is six.

Weโre asked to work out the length
of the line segment ๐ธ๐ด. So we need to recall the
relationship that exists between the lengths of the line segments of intersecting
chords. We remember that โif two chords
intersect in a circle, then the products of the lengths of the chord segments are
equal.โ

Our two chords intersect inside the
circle at the point ๐ธ. So we have that the product of the
lengths of the chord segments of the orange chord, thatโs ๐ธ๐ด multiplied by ๐ธ๐ต,
is equal to the product of the length of the chord segment of the pink cord. Thatโs ๐ธ๐ถ multiplied by ๐ธ๐ท. We know the lengths of the chord
segments ๐ธ๐ต, ๐ธ๐ถ, and ๐ธ๐ท. So we can substitute their values
into this equation given that the length of the chord segment ๐ธ๐ด multiplied by six
is equal to four multiplied by 15.

Dividing both sides of this
equation through by six gives a calculation that we can use to find the length of
the chord segment ๐ธ๐ด. Itโs equal to four multiplied by 15
over six. Four multiplied by 15 is equal to
60. And 60 divided by six is equal to
10.

Using then the relationship between
the lengths of chord segments for chords which intersect inside a circle, we found
that the length of the chord segment ๐ธ๐ด is 10. Thereโre no units for this as there
were no units for the original measurements we were given for the other three chord
segments.