The vertices of a kite have coordinates (14,2), (10,-4), (6,2), and (10,y). If the kite has an area of 64 square units, what is the value of y?
To find the value of y, we need to use the formula for the area of a kite. The area of a kite is calculated by taking half of the product of the lengths of its diagonals.
In this case, we don't know the length of the diagonals directly, but we can calculate them using the given coordinates.
The diagonal of a kite divides it into two congruent right triangles. Let's call one of these triangles ABC, where A and C are the given coordinates (14, 2) and (6, 2). To find the length of the diagonal AC, we can use the distance formula:
Distance AC = √((x₂ - x₁)² + (y₂ - y₁)²)
Plugging in the coordinates, we get:
Distance AC = √((6 - 14)² + (2 - 2)²)
= √((-8)² + 0²)
= √(64 + 0)
= √64
= 8
Similarly, we can calculate the length of the other diagonal by considering the triangle ADC, where D is the coordinate (10, y).
Distance AD = √((10 - 14)² + (y - 2)²)
= √((-4)² + (y - 2)²)
= √(16 + (y - 2)²)
= √(y² - 4y + 4 + 16)
= √(y² - 4y + 20)
Now, we have the lengths of the diagonals: AC = 8 and AD = √(y² - 4y + 20).
Since the area of the kite is given as 64 square units, we can set up the following equation:
Area = 1/2 * (AC * AD)
64 = 1/2 * (8 * √(y² - 4y + 20))
Simplifying the equation:
128 = 8√(y² - 4y + 20)
Dividing both sides of the equation by 8:
16 = √(y² - 4y + 20)
Squaring both sides of the equation to eliminate the square root:
256 = y² - 4y + 20
Rearranging the equation:
y² - 4y - 236 = 0
Now we have a quadratic equation in terms of y. To solve for y, we can either factor the quadratic or use the quadratic formula.
Using the quadratic formula:
y = (-b ± √(b² - 4ac)) / (2a)
For this equation, a = 1, b = -4, and c = -236. Plugging in these values:
y = (-(-4) ± √((-4)² - 4(1)(-236))) / (2 * 1)
= (4 ± √(16 + 944)) / 2
= (4 ± √(960)) / 2
= (4 ± 8√(15)) / 2
= 2 ± 4√(15)
Therefore, the two possible values for y are 2 + 4√(15) and 2 - 4√(15).