**Finding the Area of an Isosceles Triangle with Side Lengths 'b' and 'a'**

To find the area of an isosceles triangle, we need to know the lengths of both of its equal sides. In this case, we are given that one side of the triangle is 'b' units long, and the other two sides are equal in length.

**1. Understanding the Isosceles Triangle**

An isosceles triangle is a type of triangle that has two sides of equal length and two equal angles opposite those sides. The third side, called the base, can have a different length. In this case, the base of the triangle has a length of 'a' units.

**2. Drawing the Isosceles Triangle**

To better visualize the triangle, draw an isosceles triangle with two sides of length 'b' and the base of length 'a'. Label the triangle's vertices as A, B, and C, with A being the vertex opposite the base.

```
B
/ \
/ \
/ \
/ \
A_________C
```

**3. Height of the Triangle**

To find the area of the triangle, we need to find its height. The height of a triangle is the perpendicular distance from the base to the opposite vertex.

In this case, draw a perpendicular line from vertex A to the base line BC. Let's label the point where the perpendicular line intersects the base as D.

```
B
/ \
/ \
/ \
/ D \
A_________C
```

**4. Breaking the Triangle into Two Right-Angled Triangles**

Now, we have two right-angled triangles, ABD and ADC. Both of these triangles have a base of length 'b' and a height of length 'h'. Since the triangle is isosceles, the angles at B and C are equal, making the angles at D also equal.

```
B
/|\
/ | \
/ | \
/ | \
A____|____C
D
```

**5. The Pythagorean Theorem**

We can use the Pythagorean theorem to find the height of the triangle. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In triangle ABD, the hypotenuse is 'a', and the other two sides are 'b' (base) and 'h' (height).

Using the Pythagorean theorem, we have: a^2 = b^2 + h^2

Similarly, in triangle ADC, the hypotenuse is 'a', and the other two sides are 'b' (base) and 'h' (height).

Again, using the Pythagorean theorem, we have: a^2 = b^2 + h^2

Since both equations are the same (a^2 = b^2 + h^2), we can equate them to find the value of 'h'.

b^2 + h^2 = b^2 + h^2

Therefore, h = 0

**6. The Base of the Triangle**

Since the height of the triangle is zero, it means that