Question

The perimeter of an isosceles triangle is 36. One side is 10. What are the possible lengths of the base? Find the area of the triangle.

Answer 1

Answer:

$\boxed{\textbf{10 and 16; 60}}$

Step-by-step explanation:

1. Length of base

There are two possibilities

(a) Base = 10

Then we have a triangle as in Fig. 1.

$\begin{array}{rcl}p & = & 2x + 10\\36 & = & 2x + 10\\26 & = & 2x\\x & = & \mathbf{13}\\\end{array}$

(b) Side = 10

Then we have a triangle as in Fig. 2.

$\begin{array}{rcl}p & = & x + 20\\36 & = & x + 20\\x & = & \mathbf{16}\\\end{array}\\\text{The possible lengths of the base are \boxed{\textbf{10 and 16}} }$

2. Area of triangle

One way to find the area of the triangle is to use Heron's formula:

$A = \sqrt{s(s - a)(s - b)(s - c)}$

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

$\begin{array}{rcl}A & = & \sqrt{18(18 - 13)(18 - 13)(18 - 10)}\\& = & \sqrt{18\times 5 \times 5 \times 8}\\& = & \sqrt{3600}\\& = & \mathbf{60}\\\end{array}\\\text{The area of each triangle is \boxed{\mathbf{60}} }$