In the diagram, $\triangle A B C$ is isosceles. $M$ is on $B C$ so that $B M=M C$. If the perimeter of $\triangle A B C$ is 64 and the perimeter of $\triangle A B M$ is 40 , what is the length of $A M$ ?
(A) 10
(B) 8
(C) 16
(D) 12
(E) 24

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-013.jpg?height=209&width=463&top_left_y=2308&top_left_x=1232)

Grade 7
The perimeter of $\triangle A B C$ is equal to $A B+B C+C A$ or $A B+B M+M C+C A$.

Since $A B=C A$ and $B M=M C$, then half of the perimeter of $\triangle A B C$ is equal to $A B+B M$.

The perimeter of $\triangle A B C$ is 64 and so $A B+B M=\frac{1}{2} \times 64=32$.

The perimeter of $\triangle A B M$ is 40 , and so $A M+A B+B M=40$.

Since $A B+B M=32$, then $A M=40-32=8$.

ANswer: (B)