Answer:
[tex]\boxed{\textbf{10 and 16; 60}}[/tex]
Step-by-step explanation:
1. Length of base
There are two possibilities
(a) Base = 10
Then we have a triangle as in Fig. 1.
[tex]\begin{array}{rcl}p & = & 2x + 10\\36 & = & 2x + 10\\26 & = & 2x\\x & = & \mathbf{13}\\\end{array}[/tex]
(b) Side = 10
Then we have a triangle as in Fig. 2.
[tex]\begin{array}{rcl}p & = & x + 20\\36 & = & x + 20\\x & = & \mathbf{16}\\\end{array}\\\text{The possible lengths of the base are $\boxed{\textbf{10 and 16}}$}[/tex]
2. Area of triangle
One way to find the area of the triangle is to use Heron's formula:
[tex]A = \sqrt{s(s - a)(s - b)(s - c)}[/tex]
where s is the semiperimeter.
Each triangle has the same perimeter, so it also has the same semiperimeter and therefore the same area.
s = p/2 = 36/2 = 18
[tex]\begin{array}{rcl}A & = & \sqrt{18(18 - 13)(18 - 13)(18 - 10)}\\& = & \sqrt{18\times 5 \times 5 \times 8}\\& = & \sqrt{3600}\\& = & \mathbf{60}\\\end{array}\\\text{The area of each triangle is $\boxed{\mathbf{60}}$}[/tex]
