A quadrilateral contains two equal sides measuring 12 cm each with an included right angle. If the measure of the third side is 8 cm and the angle opposite the right angle is 120 degrees, find the fourth side and the area of the quadrilateral.

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 * (π radians / 180 degrees) = (2π / 3) radians

Area2 = (1/2) * (12 cm) * (12 cm) * sin(2π / 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.

To find the fourth side of the quadrilateral, we can use the Pythagorean theorem since we know it contains a right angle and two equal sides.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In our case, the two equal sides are 12 cm each, and the third side is 8 cm. Let's call the fourth side "x." We can set up the equation:

(12 cm)^2 + (8 cm)^2 = x^2

Simplifying:
144 cm^2 + 64 cm^2 = x^2
208 cm^2 = x^2

To find x, we need to find the square root of both sides of the equation:

√(208 cm^2) = √(x^2)

√(208 cm^2) = x

The exact value of the square root of 208 is not a whole number, so the fourth side of the quadrilateral is approximately 14.422 cm (rounded to three decimal places).

To find the area of the quadrilateral, we can divide it into two right triangles and a rectangle. The area of a right triangle is given by the formula: (1/2) * base * height, and the area of a rectangle is given by the formula: length * width.

The two right triangles formed have a base of 8 cm and a height of 12 cm. Their combined area is:
(1/2) * 8 cm * 12 cm = 48 cm^2

The rectangle has a width of 8 cm and a length of 12 cm. Its area is:
8 cm * 12 cm = 96 cm^2

To find the total area of the quadrilateral, we add the areas of the triangles and the rectangle:
Total area = 48 cm^2 + 96 cm^2 = 144 cm^2

Therefore, the area of the quadrilateral is 144 square cm.