Find the perimeter of the triangle formed by the points $(3, 5), (4, 8)$ and $(5, 6)$.

To find the perimeter of the triangle formed by the points $A(3,5)$,

$B(4,8)$ and

$C(5,6)$, we must find the distance between these three points that can be determined as follows:

$AB=\sqrt { { (4-3) }^{ 2 }+{ (8-5) }^{ 2 } } =\sqrt { { 1 }^{ 2 }+{ 3 }^{ 2 } } =\sqrt { 1+9 } =\sqrt { 10 }$

$BC=\sqrt { { (5-4) }^{ 2 }+{ (6-8) }^{ 2 } } =\sqrt { { 1 }^{ 2 }+{ 2 }^{ 2 } } =\sqrt { 1+4 } =\sqrt { 5 }$

$AC=\sqrt { { (5-3) }^{ 2 }+{ (6-5) }^{ 2 } } =\sqrt { { 2 }^{ 2 }+{ 1 }^{ 2 } } =\sqrt { 4+1 } =\sqrt { 5 }$

Now the perimeter is $AB+BC+AC$ that is

$AB+BC+AC=\sqrt { 10 } +\sqrt { 5 } +\sqrt { 5 } =\sqrt { 10 } +2\sqrt { 5 } =\sqrt { 5 } (2+\sqrt { 2 } )$

Hence the perimeter is

$\sqrt { 5 } (2+\sqrt { 2 } )$.