### Video Transcript

In a circle of center π, π΄π΅ is
equal to 35 centimeters, πΆπ΅ is equal to 25 centimeters, and π΄πΆ is equal to 40
centimeters. Given that line segment ππ· is
perpendicular to line segment π΅πΆ and line segment ππΈ is perpendicular to line
segment π΄πΆ, find the perimeter of triangle πΆπ·πΈ.

We are given in the question the
length of the three sides of the triangle πΆπ΅π΄. We know that π΄π΅ is 35
centimeters, πΆπ΅ is 25 centimeters, and π΄πΆ is 40 centimeters. We have been asked to calculate the
perimeter of triangle πΆπ·πΈ. We will do this by firstly proving
that triangles πΆπ΅π΄ and πΆπ·πΈ are similar using the chord bisector theorem. We notice from the diagram that the
line segments ππΈ and ππ· both pass through π and meet the chords π΄πΆ and πΆπ΅
at right angles.

The chord bisector theorem states
that if we have a circle with center π containing a chord π΅πΆ, then the straight
line that passes through π and is perpendicular to π΅πΆ also bisects π΅πΆ. In our diagram, this means that the
length of π΄πΈ is equal to the length πΈπΆ and the length πΆπ· is equal to the
length π·π΅.

It is also clear from the diagram
that π΄πΆ is equal to two multiplied by πΈπΆ and πΆπ΅ is equal to two multiplied by
πΆπ·. As the two triangles πΆπ΅π΄ and
πΆπ·πΈ also share the angle πΆ, we have two corresponding sides in proportion and
the angle between the two sides is congruent. This proves that the two triangles
are similar. And in fact triangle πΆπ΅π΄ is
larger than triangle πΆπ·πΈ by a scale factor of two, as the lengths of the
corresponding sides are twice as long. Side π΄πΆ is equal to two
multiplied by side πΈπΆ, πΆπ΅ is equal to two πΆπ·, and π΄π΅ is equal to two
multiplied by πΈπ·.

We can calculate the perimeter of
triangle πΆπ΅π΄ by adding 40, 35, and 25. This is equal to 100
centimeters. The perimeter of triangle πΆπ·πΈ
will therefore be equal to half of this. This is equal to 50
centimeters.