In the diagram, triangles $A B C$ and $C B D$ are isosceles. The perimeter of $\triangle C B D$ is 19 , the perimeter of $\triangle A B C$ is 20 , and the length of $B D$ is 7 . What is the length of $A B$ ?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-098.jpg?height=331&width=548&top_left_y=1881&top_left_x=1206)
In $\triangle A B C, \angle A B C=\angle B A C$, so $A C=B C$.

In $\triangle B C D, \angle C B D=\angle C D B$, so $C D=B C$.

Since the perimeter of $\triangle C B D$ is 19 and $B D=7$, then $7+B C+C D=19$ or $2(B C)=12$ or $B C=6$.

Since the perimeter of $\triangle A B C$ is $20, B C=6$, and $A C=B C$, then $A B+6+6=20$ or $A B=8$.

ANSWER: (D)

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