### Video Transcript

In the figure below, if ππ΄ is
equal to 17.2 centimeters and π΄π΅ is equal to 27.6 centimeters, find the length of
the line segment ππΆ and the area of triangle π΄π·π΅ to the nearest tenth.

Since π is the center of the
circle and the line segment ππ· bisects the chord π΄π΅ at πΆ, we can apply the
chord bisector theorem. This states that if we have a
circle with center π containing a chord π΄π΅, then the straight line that passes
through π and bisects π΄π΅ is perpendicular to π΄π΅. This means that the measure of
angle ππΆπ΄ is 90 degrees. Since the chord π΄π΅ has length
27.6 centimeters and we know that πΆ bisects this chord, π΄πΆ is equal to 27.6
divided by two. This is equal to 13.8
centimeters.

We are also told that the radius
ππ΄ is equal to 17.2 centimeters. We can therefore use the
Pythagorean theorem in the right triangle ππΆπ΄ such that ππΆ is equal to the
square root of 17.2 squared minus 13.8 squared. This is equal to 10.266 and so
on. And rounding to the nearest tenth,
the line segment ππΆ is equal to 10.3 centimeters.

The second part of our question
asks us to calculate the area of triangle π΄π·π΅. Clearing some space, we recall that
the area of any triangle is equal to the length of its base multiplied by the length
of its perpendicular height divided by two. We know that the base of our
triangle π΄π΅ is equal to 27.6 centimeters. The perpendicular height πΆπ· will
be equal to the length of ππ· minus the length of ππΆ. ππ· is the radius of the circle,
and we know this is equal to 17.2 centimeters. Whilst we could use 10.3
centimeters for ππΆ, it is better for accuracy to use the nonrounded version: ππΆ
is equal to 10.266 and so on. Subtracting this from 17.2, we see
that the length of πΆπ· is 6.933 and so on centimeters.

We can now calculate the area of
triangle π΄π·π΅ by multiplying this by 27.6 centimeters and then dividing by
two. This is equal to 95.683 and so
on. Once again, we need to round to the
nearest tenth, giving us an answer of 95.7 square centimeters.