### Video Transcript

If 𝑏 is the middle proportion
between 𝑎 and 𝑐, then which of the following is equal to 𝑎 squared plus 𝑏
squared over 𝑏 squared plus 𝑐 squared? Is it option (A) 𝑐 over 𝑎, option
(B) 𝑎 over 𝑐, option (C) two 𝑐 over 𝑎, or option (D) two 𝑎 over 𝑐?

We will begin by recalling the
definition of the middle proportion. We know that if 𝑏 is the middle
proportion between 𝑎 and 𝑐, then 𝑎 over 𝑏 is equal to 𝑏 over 𝑐. In this question, we are given an
expression involving 𝑎 squared, 𝑏 squared, and 𝑐 squared, and we need to decide
which of the four options given are equal to this. We notice that none of these
options contain 𝑏. The equation 𝑎 over 𝑏 is equal to
𝑏 over 𝑐 can be rewritten as 𝑎𝑐 is equal to 𝑏 squared.

We can now substitute the
expression for 𝑏 squared into our expression 𝑎 squared plus 𝑏 squared over 𝑏
squared plus 𝑐 squared. This becomes 𝑎 squared plus 𝑎𝑐
over 𝑎𝑐 plus 𝑐 squared. The two terms on the numerator have
a common factor of 𝑎. We can therefore factor that this
out, giving us 𝑎 multiplied by 𝑎 plus 𝑐 since 𝑎 multiplied by 𝑎 is 𝑎 squared
and 𝑎 multiplied by 𝑐 is 𝑎𝑐. The two terms on the denominator
have a common factor of 𝑐. Factoring this out gives us 𝑐
multiplied by 𝑎 plus 𝑐. Since both the numerator and
denominator have a common factor of 𝑎 plus 𝑐, we can cancel this, leaving us with
a simplified expression, 𝑎 over 𝑐.

If 𝑏 is the middle proportion
between 𝑎 and 𝑐, then 𝑎 squared plus 𝑏 squared over 𝑏 squared plus 𝑐 squared
is equal to 𝑎 over 𝑐. Of the four options listed, the
correct answer is option (B).