Answer: Area = = 38.5 - Perimeter - 29.6819.
Explanation:
To find the perimeter and area of a triangle with vertices L(5,5), M(-2,-4), and N(5,-6), we can use the distance formula and the formula for the area of a triangle.
Perimeter:
The perimeter of a triangle is the sum of the lengths of its sides. We can find the lengths of each side using the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, we can calculate the lengths of LM, MN, and NL:
LM = √((-2 - 5)^2 + (-4 - 5)^2)
= √((-7)^2 + (-9)^2)
= √(49 + 81)
= √130
MN = √((5 - (-2))^2 + (-6 - (-4))^2)
= √((7)^2 + (-2)^2)
= √(49 + 4)
= √53
NL = √((5 - 5)^2 + (-6 - 5)^2)
= √((0)^2 + (-11)^2)
= √(0 + 121)
= √121
= 11
The perimeter of the triangle is the sum of these three sides:
Perimeter = LM + MN + NL
= √130 + √53 + 11 = 29.6819
Area:
To calculate the area of the triangle, we can use the formula for the area of a triangle using the coordinates of its vertices. The formula is given by:
Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
Using the coordinates of L(5,5), M(-2,-4), and N(5,-6), we can calculate the area:
Area = 0.5 * |5(-4 - (-6)) + (-2)(-6 - 5) + 5(5 - (-4))|
= 0.5 * |5(2) + (-2)(-11) + 5(9)|
= 0.5 * |10 + 22 + 45|
= 0.5 * |77|
= 38.5
Therefore, the perimeter of the triangle is given by √130 + √53 + 11 = 29.6819 units, and the area of the triangle is 38.5 square units.