To find the fourth side of the quadrilateral, we can use the Pythagorean theorem. In a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, since we have a right angle and two equal sides of 12 cm each, we can consider the triangle formed by these sides and the unknown fourth side. Let's call the fourth side "x".
Using the Pythagorean theorem, we can set up the equation:
x^2 = (12 cm)^2 + (12 cm)^2
Simplifying, we have:
x^2 = 144 cm^2 + 144 cm^2
x^2 = 288 cm^2
Taking the square root of both sides, we find:
x = sqrt(288) cm
x ≈ 16.97 cm (rounded to two decimal places)
So, the length of the fourth side of the quadrilateral is approximately 16.97 cm.
Next, let's find the area of the quadrilateral. To do this, we can divide it into two triangles and find the sum of their areas.
The first triangle is a right triangle with one side measuring 8 cm and the other side measuring 12 cm. We can find its area using the formula:
Area = (base * height) / 2
Area1 = (8 cm * 12 cm) / 2
Area1 = 96 cm^2
The second triangle is an isosceles triangle with two equal sides measuring 12 cm and an angle opposite one of the equal sides measuring 120 degrees. We can find its area using the formula:
Area2 = (1/2) * (12 cm) * (12 cm) * sin(120 degrees)
To calculate the sine of 120 degrees, we need to convert it to radians:
120 degrees * (Ï€ radians / 180 degrees) = (2Ï€ / 3) radians
Area2 = (1/2) * (12 cm) * (12 cm) * sin(2Ï€ / 3)
Area2 = 72 * (√3 / 2) cm^2
Area2 ≈ 62.35 cm^2 (rounded to two decimal places)
Finally, the area of the quadrilateral is the sum of the areas of the two triangles:
Area = Area1 + Area2
Area ≈ 96 cm^2 + 62.35 cm^2
Area ≈ 158.35 cm^2 (rounded to two decimal places)
Therefore, the area of the quadrilateral is approximately 158.35 square centimeters.