Perimeter of the Isosceles Triangle

Let's assume that the length of the two equal sides of the isosceles triangle is '5x' and '4x' respectively, and the length of the base is 'y'.

The perimeter of a triangle is the sum of all its sides. In this case, the perimeter is given as 14 cm. Therefore, we can write the equation as:

5x + 4x + y = 14

Simplifying the equation, we get:

9x + y = 14

Ratio of the Letters at the Base

The ratio of the letters at the base is given as 5:4. This means that the length of the base can be expressed as:

y = k(5 + 4)

Where 'k' is a constant.

Simplifying the equation, we get:

y = 9k

Finding the Length of the Base

From the above equation, we know that y = 9k. Substituting this value into the perimeter equation, we get:

9x + 9k = 14

Dividing both sides of the equation by 9, we get:

x + k = 14/9

Calculating the Area of the Triangle

The area of a triangle can be calculated using the formula:

Area = (base * height) / 2

In this case, the base is '9k' and the height can be determined using the Pythagorean theorem, since the isosceles triangle is also a right triangle.

Using the Pythagorean theorem, we can find the height 'h' as:

h^2 = (4x)^2 - (2.5x)^2

Simplifying the equation, we get:

h^2 = 16x^2 - 6.25x^2
h^2 = 9.75x^2

Taking the square root of both sides, we get:

h = sqrt(9.75)x

Substituting the values of 'h' and 'y' into the area formula, we have:

Area = (9k * sqrt(9.75)x) / 2

Simplifying the equation, we get:

Area = (9k * sqrt(9.75)x) / 2

Therefore, the area of the isosceles triangle is given by the above expression.