This is a two-part item. A garden in a park is in the shape of a right triangle. A map of the garden is drawn on a coordinate grid where each unit represents 1 meter. The vertices of the garden are located at (−4, 1), (0,−2), and (3, 2).
Part A: What is the area of the garden in square meters?
square meters
Part B: What is the perimeter of the garden in meters? Round to the hundredths, if necessary.
7.07
10
12.07
17.07
Part A:
The formula for finding the area of a triangle given the coordinates of its vertices is:
Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
Plugging in the coordinates of the vertices:
Area = 1/2 * |-4(-2 - 2) + 0(2 - 1) + 3(1 + 2)|
Area = 1/2 * |-4(-4) + 0(1) + 3(3)|
Area = 1/2 * (16 + 0 + 9)
Area = 1/2 * 25
Area = 12.5 square meters
Therefore, the area of the garden is 12.5 square meters.
Part B:
To find the perimeter, we need to find the distance between each pair of vertices and then sum them up:
Distance between (-4, 1) and (0, -2):
√((-4 - 0)^2 + (1 - (-2))^2) = √(16 + 9) = √25 = 5 meters
Distance between (0, -2) and (3, 2):
√((0 - 3)^2 + (-2 - 2)^2) = √(9 + 16) = √25 = 5 meters
Distance between (3, 2) and (-4, 1):
√((3 - (-4))^2 + (2 - 1)^2) = √(49 + 1) = √50 = 7.07 meters
Summing up the distances:
5 + 5 + 7.07 = 17.07 meters
Therefore, the perimeter of the garden is 17.07 meters.