Video Transcript
In the figure shown, the circle has
a radius of 12 centimeters. 𝐴𝐵 is equal to 12 centimeters,
and 𝐴𝐶 is equal to 35 centimeters. Determine the distance from line
segment 𝐵𝐶 to the center of the circle 𝑀 and the length of line segment 𝐴𝐷,
rounding your answers to the nearest tenth.
We’re going to begin by finding the
distance from line segment 𝐵𝐶 to the center of the circle 𝑀. We might recall that the shortest
distance from a point to a line is the length of the perpendicular from that point
to the line. And so we construct this
perpendicular from point 𝑀 to the line segment 𝐵𝐶. In fact, since 𝑀 is the center of
the circle and 𝐵𝐶 is a chord, we can say that this perpendicular is the
perpendicular line bisector of 𝐵𝐶. So defining the point where this
perpendicular meets the line segment 𝐵𝐶 as 𝐸, we can say that 𝐵𝐸 must be equal
to 𝐸𝐶.
Next, we’re going to use the fact
that the radius of the circle is 12 centimeters. The radius, of course, is the line
segment that joins the point of the center of the circle to any point on its the
circumference. So we can say that 𝑀𝐵 is 12
centimeters.
Next, we apply the fact that 𝐴𝐵
is equal to 12 centimeters and 𝐴𝐶 is equal to 35 centimeters. Since we can think of line segment
𝐴𝐶 as the sum of line segments 𝐴𝐵 and 𝐵𝐶, we can say that 35 is equal to 12
plus 𝐵𝐶 and we can find the length of 𝐵𝐶 by subtracting 12 from both sides of
this equation. 35 minus 12 is 23. So 𝐵𝐶 is 23 centimeters in
length.
But remember, we said that the line
segment 𝑀𝐸 is the perpendicular bisector for the line segment 𝐵𝐶. So 𝐵𝐸 must be half of 𝐵𝐶, that
is, 23 divided by two or 23 over two centimeters. We now note that we have a right
triangle 𝑀𝐸𝐵 for which we know two of its sides. We can therefore use the
Pythagorean theorem to find the length of the side 𝑀𝐸. Let’s call that 𝑥 or 𝑥
centimeters.
Substituting what we know about
this triangle into the Pythagorean theorem, and we find that 12 squared equals 𝑥
squared plus 23 over two squared. Then we make 𝑥 squared the subject
by subtracting 23 over two squared from both sides. 12 squared minus 23 over two
squared is 47 over four. To find the length that we’re
interested in, 𝑥, we’re going to find the positive square root of 47 over four. And that’s equal to 3.427 and so
on. Correct the nearest tenth, we find
that’s equal to 3.4 centimeters.
We now move on to the second part
of this question. And that asks us to find the length
of line segment 𝐴𝐷. And we observed that line segment
𝐴𝐷 is, in fact, a tangent segment, whilst the line 𝐴𝐶 is a secant segment. This means we can use a special
version of the intersecting secants theorem. And that’s called the tangent
secant theorem. In the case of our circle, it tells
us that the product of the lengths of line segments 𝐴𝐵 and 𝐴𝐶 is equal to the
square of the length of line segment 𝐴𝐷.
Now we’re given that 𝐴𝐵 is 12
centimeters, whilst 𝐴𝐶 is 35. So 12 times 35 is equal to 𝐴𝐷
squared, or 𝐴𝐷 squared is equal to 420. We’ll solve this equation by
finding the square root of 420. That gives us that 𝐴𝐷 is 20.493,
which correct the nearest tenth is 20.5 centimeters. The distance from line segment 𝐵𝐶
to the center of the circle 𝑀 is 3.4 centimeters, and the length of line segment
𝐴𝐷 is 20.5 centimeters.
