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.
d^4=100/6
As usual, draw a figure. If we start at the right angle and label the vertices clockwise, then we see that BD is the hypotenuse of an isosceles right triangle, with length 12√2.
If the 4th side is x, then we have from the law of cosines,
(12√2)^2 = 8^2 + x^2 - 2(8)(x)cos120°
or,
x^2 + 8x - 224 = 0
x = 4(√15-1) = 11.5
Now you can split the area into a rectangle and two triangles and find the area.
To find the length of the fourth side, we can use the Pythagorean theorem. In a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Let's label the fourth side as "x".
Using the Pythagorean theorem, we have:
x^2 = (12 cm)^2 + (8 cm)^2
x^2 = 144 cm^2 + 64 cm^2
x^2 = 208 cm^2
Taking the square root of both sides:
x = √208 cm
x ≈ 14.42 cm (rounded to two decimal places)
Therefore, the length of the fourth side is approximately 14.42 cm.
To find the area of the quadrilateral, we need to divide it into two right-angled triangles.
Area of a triangle = base × height / 2
The base of each triangle is 12 cm, and the height is 8 cm.
Area of each triangle = (12 cm × 8 cm) / 2 = 96 cm²
Since the quadrilateral is made up of two triangles, the total area is twice the area of one triangle.
Total area of the quadrilateral = 2 × 96 cm² = 192 cm²
Therefore, the area of the quadrilateral is 192 cm².
To find the fourth side, we can use the Pythagorean theorem, since we have a right angle.
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 this case, the two equal sides measuring 12 cm each are the legs of the right triangle, and the unknown fourth side is the hypotenuse.
Using the Pythagorean theorem, we can calculate the length of the fourth side:
Let's call the fourth side 'x'.
Using the Pythagorean theorem: (12 cm)^2 + (8 cm)^2 = x^2.
So, 144 cm^2 + 64 cm^2 = x^2.
Simplifying, we get: 208 cm^2 = x^2.
To solve for x, we take the square root of both sides: sqrt(208 cm^2) = sqrt(x^2).
Therefore, x = sqrt(208) cm.
Now, let's calculate the area of the quadrilateral.
Since we have a right angle and one side measuring 8 cm, we can split the quadrilateral into two right triangles.
Let's call the height of the quadrilateral 'h'.
The area of each right triangle is given by the formula: (1/2) * base * height.
In our case, the base of each right triangle is 8 cm, and the height is h.
Therefore, the area of each right triangle is: (1/2) * 8 cm * h.
The total area of the quadrilateral is the sum of the areas of the two right triangles.
So, the total area = 2 * (1/2) * 8 cm * h = 8 cm * h.
Now, we need to find the height 'h' of the quadrilateral.
To do this, we can use trigonometry. Since we know the angle opposite the right angle is 120 degrees, we can use the sine function.
The sine function relates the opposite side to the hypotenuse in a right triangle.
Sine of an angle = opposite side / hypotenuse.
In our case, the opposite side is 'h' and the hypotenuse is sqrt(208) cm.
So, sin(120 degrees) = h / sqrt(208) cm.
To find 'h', we rearrange the equation:
h = sin(120 degrees) * sqrt(208) cm.
Now, we can substitute this value of 'h' into the equation for the area:
Area = 8 cm * h = 8 cm * sin(120 degrees) * sqrt(208) cm.
Therefore, the fourth side of the quadrilateral is sqrt(208) cm, and the area of the quadrilateral is 8 cm * sin(120 degrees) * sqrt(208) cm.