a parallelogram has the vertices (-1,2), (4,4), (2,-1), and (-3,-3). determine what type of parallelogram. find the perimeter and area. i found the parallelogram part and its a rhombus but i cant find the perimeter and area
You must have found the length of each side to determine it was a rhomus
I found each side to be √29
so just take 4√29 for the perimeter.
the area of a rhomus = (1/2)product of their diagonals
so just find the length of the diagonals and go from there.
To find the perimeter and area of a parallelogram, regardless of its type, you need to know the lengths of its sides and the lengths of its diagonals (if necessary).
In this case, since you have identified the given parallelogram as a rhombus, we can conclude that all the sides are equal, and the diagonals are perpendicular bisectors of each other.
Let's calculate the lengths of the sides first:
Side AB:
Using the distance formula:
AB = √((x2 - x1)^2 + (y2 - y1)^2)
= √((4 - (-1))^2 + (4 - 2)^2)
= √((5)^2 + (2)^2)
= √(25 + 4)
= √29
Similarly, you can calculate the length of the other three sides:
BC = √((2 - 4)^2 + (-1 - 4)^2) = √5
CD = √((-3 - 2)^2 + (-3 - (-1))^2) = √17
DA = √((-1 - (-3))^2 + (2 - (-3))^2) = √45
Since the opposite sides of a rhombus are equal, we can conclude that all the side lengths are the same: AB = BC = CD = DA = √29.
Now, to find the perimeter of the rhombus, you simply add up the lengths of all four sides:
Perimeter = AB + BC + CD + DA = √29 + √5 + √17 + √45
Next, let's find the area of the rhombus. Since the diagonals bisect each other perpendicularly, they divide the rhombus into four congruent right triangles. The formula for the area of a rhombus in terms of the length of its diagonals (d1 and d2) is:
Area = (d1 * d2) / 2
To find the lengths of the diagonals, use the distance formula again:
Diagonal AC:
AC = √((-3 - (-1))^2 + (-3 - 2)^2) = √40
Diagonal BD:
BD = √((4 - 2)^2 + (4 - (-1))^2) = √26
Now we can calculate the area of the rhombus:
Area = (AC * BD) / 2 = (√40 * √26) / 2
To simplify the area further, combine the square roots:
Area = (√(40 * 26)) / 2 = (√1040) / 2 = (√(4 * 260)) / 2 = (2√260) / 2 = √260
Thus, the perimeter of the given rhombus is √29 + √5 + √17 + √45, and the area is √260.