Answer:
25.6 units
Step-by-step explanation:
From the figure we can infer that our triangle has vertices A = (-5, 4), B = (1, 4), and C = (3, -4).
First thing we are doing is find the lengths of AB, BC, and AC using the distance formula:
![d=\sqrt{(x_2-x_1)^{2} +(y_2-y_1)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28x_2-x_1%29%5E%7B2%7D%20%2B%28y_2-y_1%29%5E%7B2%7D%7D)
where
are the coordinates of the first point
are the coordinates of the second point
- For AB:
![d=\sqrt{[1-(-5)]^{2}+(4-4)^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%5B1-%28-5%29%5D%5E%7B2%7D%2B%284-4%29%5E2%7D)
![d=\sqrt{(1+5)^{2}+(0)^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%281%2B5%29%5E%7B2%7D%2B%280%29%5E2%7D)
![d=\sqrt{(6)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%286%29%5E%7B2%7D%7D)
![d=6](https://tex.z-dn.net/?f=d%3D6)
- For BC:
![d=\sqrt{(3-1)^{2} +(-4-4)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%283-1%29%5E%7B2%7D%20%2B%28-4-4%29%5E%7B2%7D%7D)
![d=\sqrt{(2)^{2} +(-8)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%282%29%5E%7B2%7D%20%2B%28-8%29%5E%7B2%7D%7D)
![d=\sqrt{4+64}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B4%2B64%7D)
![d=\sqrt{68}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B68%7D)
![d=8.24](https://tex.z-dn.net/?f=d%3D8.24)
- For AC:
![d=\sqrt{[3-(-5)]^{2} +(-4-4)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%5B3-%28-5%29%5D%5E%7B2%7D%20%2B%28-4-4%29%5E%7B2%7D%7D)
![d=\sqrt{(3+5)^{2} +(-8)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%283%2B5%29%5E%7B2%7D%20%2B%28-8%29%5E%7B2%7D%7D)
![d=\sqrt{(8)^{2} +64}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%288%29%5E%7B2%7D%20%2B64%7D)
![d=\sqrt{64+64}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B64%2B64%7D)
![d=\sqrt{128}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B128%7D)
![d=11.31](https://tex.z-dn.net/?f=d%3D11.31)
Next, now that we have our lengths, we can add them to find the perimeter of our triangle:
![p=AB+BC+AC](https://tex.z-dn.net/?f=p%3DAB%2BBC%2BAC)
![p=6+8.24+11.31](https://tex.z-dn.net/?f=p%3D6%2B8.24%2B11.31)
![p=25.55](https://tex.z-dn.net/?f=p%3D25.55)
![p=25.6](https://tex.z-dn.net/?f=p%3D25.6)
We can conclude that the perimeter of the triangle shown in the figure is 25.6 units.