Question

during recess, three friends decided to play a game of catch in a field. if their locations on the field are drawn on a coordinate plane, where the x - and y - values represent the length, in feet, the friends locations are as follows.
kylie is located at (3, 5).
jackson is located at (11, - 3).
hailey is located at (3, - 11).
what is the area of the field between the three friends?
a. 128\sqrt{2} square feet
b. 128 square feet
c. 64 square feet
d. (16\sqrt{2}+16) square feet

Explanation:

Step1: Determine the base and height

Kylie is at \((3,5)\) and Hailey is at \((3, - 11)\). The distance between them (vertical side, height \(h\)) is calculated using the formula for the distance between two points with the same \(x\) - coordinate: \(h=\vert y_2 - y_1\vert\).
\(h=\vert-11 - 5\vert=\vert-16\vert = 16\) feet.
Jackson is at \((11,-3)\) and Kylie is at \((3,5)\). The distance between Kylie \((3,5)\) and Jackson \((11,-3)\) (horizontal side, base \(b\)):
Since the line connecting Kylie \((3,5)\) and Hailey \((3,-11)\) is vertical (\(x = 3\)), and we can find the horizontal distance from the \(x\) - coordinates of Kylie \((x_1 = 3)\) and Jackson \((x_2=11)\). The base \(b=\vert11 - 3\vert=8\) feet.

Step2: Calculate the area of the triangle

The area of a triangle is given by the formula \(A=\frac{1}{2}\times base\times height\).
Substitute \(b = 8\) and \(h = 16\) into the formula:
\(A=\frac{1}{2}\times8\times16\)
\(A = 64\) square feet.

Answer:

C. 64 square feet