Find the perimeter and area.
a.
(1,4), (4,5), (5,2)
b.
(0,–2), (0,2), (4,0)
c.
(0, 0), (2, 0), (4,–3)
for the perimeters, just find the lengths of the sides and add them up.
The areas are a bit trickier, but
(a) once you get the sides, you will see that it is a right triangle, so the area is easy.
(b) is easy, since it is easy to see that it is isosceles, with base=4 and height=4
(c) is a little tougher, but you can use Heron's formula, giving the area as
√(((7+√13)/2)((3+√13)/2)((-3+√13)/2)((7-√13)/2)) = 3
note that the line from (0,0) to (4,-3) is 3x+4y=0. The altitude is the distance from (2,0) to that line, which is 6/5, making the area = 3, or
note that the perpendicular to 3x+4y=0 through (2,0) is y = 4/3 (x-2). The two lines intersect at (32/25,-24/25), so the altitude is 6/5, making the area = 3
What tools do you have?
Do you know how to find the distance between two points?
Do you know how to find angles of a triangle knowing the sides?
To find the perimeter and area of a given figure, we need to know the coordinates of the points that define its shape. Let's calculate the perimeter and area for each of the given figures.
a.
To find the perimeter, we need to calculate the sum of the lengths of all sides of the figure. The given coordinates are (1,4), (4,5), and (5,2).
First, we calculate the distances between each pair of consecutive points:
- Distance between (1,4) and (4,5) = sqrt((4-1)^2 + (5-4)^2) = sqrt(3^2 + 1^2) = sqrt(9 + 1) = sqrt(10).
- Distance between (4,5) and (5,2) = sqrt((5-4)^2 + (2-5)^2) = sqrt(1^2 + (-3)^2) = sqrt(1 + 9) = sqrt(10).
Next, we calculate the distance between the first and last points:
- Distance between (5,2) and (1,4) = sqrt((1-5)^2 + (4-2)^2) = sqrt((-4)^2 + 2^2) = sqrt(16 + 4) = sqrt(20) = 2 * sqrt(5).
Finally, we sum up these distances to find the perimeter:
Perimeter = Distance between (1,4) and (4,5) + Distance between (4,5) and (5,2) + Distance between (5,2) and (1,4)
= sqrt(10) + sqrt(10) + 2 * sqrt(5)
= 2 * sqrt(10) + 2 * sqrt(5)
≈ 9.90
To find the area, we need to calculate the area of the triangle formed by the given points. We can use the Shoelace Formula to find the area of any polygon given its vertices.
Area = 1/2 * abs[(x1y2 + x2y3 + x3y1) - (x2y1 + x3y2 + x1y3)]
= 1/2 * abs[(1*5 + 4*2 + 5*4) - (4*4 + 5*1 + 1*2)]
= 1/2 * abs[(5 + 8 + 20) - (16 + 5 + 2)]
= 1/2 * abs[33 - 23]
= 1/2 * abs[10]
= 5 square units
Therefore, for figure a, the perimeter is approximately 9.90 units, and the area is 5 square units.
b.
Using the same approach, we can find the perimeter and area for figure b with the coordinates (0,–2), (0,2), and (4,0).
Perimeter = Distance between (0,–2) and (0,2) + Distance between (0,2) and (4,0) + Distance between (4,0) and (0,–2)
= 2 + 2 + 2 * sqrt(2)
= 4 + 2 * sqrt(2)
≈ 6.83
Area = 1/2 * abs[(x1y2 + x2y3 + x3y1) - (x2y1 + x3y2 + x1y3)]
= 1/2 * abs[(0*2 + 0*0 + 4*(-2)) - (0*(-2) + 4*0 + 0*2)]
= 1/2 * abs[0 - 0]
= 0 square units
Therefore, for figure b, the perimeter is approximately 6.83 units, and the area is 0 square units (since it's a straight line).
c.
Using the same approach, we can find the perimeter and area for figure c with the coordinates (0, 0), (2, 0), and (4,–3).
Perimeter = Distance between (0, 0) and (2, 0) + Distance between (2, 0) and (4,–3) + Distance between (4,–3) and (0, 0)
= 2 + sqrt(13) + sqrt(25)
= 7 + sqrt(13)
≈ 10.60
Area = 1/2 * abs[(x1y2 + x2y3 + x3y1) - (x2y1 + x3y2 + x1y3)]
= 1/2 * abs[(0*0 + 2*(-3) + 4*0) - (2*0 + 4*0 + 0*(-3))]
= 1/2 * abs[0 - 0]
= 0 square units
Therefore, for figure c, the perimeter is approximately 10.60 units, and the area is 0 square units (since it's a straight line).