Answer:
![Perimeter = 38.6](https://tex.z-dn.net/?f=Perimeter%20%3D%2038.6)
Step-by-step explanation:
Given
![A\ B = 13](https://tex.z-dn.net/?f=A%5C%20B%20%3D%2013)
![B\ F = 9](https://tex.z-dn.net/?f=B%5C%20F%20%3D%209)
![F\ C = 5.6](https://tex.z-dn.net/?f=F%5C%20C%20%3D%205.6)
Required
Calculate the perimeter of A B C
To calculate the perimeter of A B C, we sum up the sides of the triangle
i.e.
![Perimeter = A B + B\ C + A\ C](https://tex.z-dn.net/?f=Perimeter%20%3D%20A%20B%20%2B%20B%5C%20C%20%2B%20A%5C%20C)
AB is given in the question already.
So, we need to calculate B F and F C
To solve for BF, we consider triangle B F C
Using Pythagoras Theorem, we have:
![B\ C^2 = B\ F^2 + F\ C^2](https://tex.z-dn.net/?f=B%5C%20C%5E2%20%3D%20B%5C%20F%5E2%20%2B%20F%5C%20C%5E2)
Substitute values for BF and FC;
![B\ C^2 = 9^2 + 5.6^2](https://tex.z-dn.net/?f=B%5C%20C%5E2%20%3D%209%5E2%20%2B%205.6%5E2)
![B\ C^2 = 81 + 31.36](https://tex.z-dn.net/?f=B%5C%20C%5E2%20%3D%2081%20%2B%2031.36)
![B\ C^2 = 112.36](https://tex.z-dn.net/?f=B%5C%20C%5E2%20%3D%20112.36)
![B\ C^2= \sqrt{112.36](https://tex.z-dn.net/?f=B%5C%20C%5E2%3D%20%5Csqrt%7B112.36)
![B\ C = 10.6](https://tex.z-dn.net/?f=B%5C%20C%20%3D%2010.6)
Next, we calculate A C.
To calculate A C, we need to calculate A F, considering triangle B F A
Using Pythagoras Theorem, we have:
![A\ B^2 = B\ F^2 + A\ F^2](https://tex.z-dn.net/?f=A%5C%20B%5E2%20%3D%20B%5C%20F%5E2%20%2B%20A%5C%20F%5E2)
Substitute values for B F and A B;
![13^2 = 9^2 + A\ F^2](https://tex.z-dn.net/?f=13%5E2%20%3D%209%5E2%20%2B%20A%5C%20F%5E2)
![169= 81 + A\ F^2](https://tex.z-dn.net/?f=169%3D%2081%20%2B%20A%5C%20F%5E2)
![A\ F^2 = 169 - 81](https://tex.z-dn.net/?f=A%5C%20F%5E2%20%3D%20169%20-%2081)
![A\ F^2 = 88](https://tex.z-dn.net/?f=A%5C%20F%5E2%20%3D%2088)
![A\ F = \sqrt{88](https://tex.z-dn.net/?f=A%5C%20F%20%3D%20%5Csqrt%7B88)
![A\ F = 9.4](https://tex.z-dn.net/?f=A%5C%20F%20%3D%209.4)
Since F lies on line A C:
![A\ C = A\ F + F\ C](https://tex.z-dn.net/?f=A%5C%20C%20%3D%20A%5C%20F%20%2B%20F%5C%20C)
![A\ C = 9.4 + 5.6](https://tex.z-dn.net/?f=A%5C%20C%20%3D%209.4%20%2B%205.6)
![A\ C = 15.0](https://tex.z-dn.net/?f=A%5C%20C%20%3D%2015.0)
Recall that:
![Perimeter = A\ B + B\ C + A\ C](https://tex.z-dn.net/?f=Perimeter%20%3D%20A%5C%20B%20%2B%20B%5C%20C%20%2B%20A%5C%20C)
![Perimeter = 13 + 10.6 + 15.0](https://tex.z-dn.net/?f=Perimeter%20%3D%2013%20%2B%2010.6%20%2B%2015.0)
![Perimeter = 38.6](https://tex.z-dn.net/?f=Perimeter%20%3D%2038.6)
<em>Hence, the perimeter is 38.6</em>