Find the area and perimeter of the figure determined by the points.(−6, 0)​, (−1, −4)​, (3, 1)

Question

Find the area and perimeter of the figure determined by the points.(−6, 0)​, (−1, −4)​, (3, 1)

Answer 1

To find the area of the figure determined by the given points, we can use the Shoelace Formula, which works for any polygon with vertices given in order.

The Shoelace Formula states that if the vertices of a polygon are given in order as (x1, y1), (x2, y2), ..., (xn, yn), then the area of the polygon is given by:

area = 1/2 * |(x1y2 + x2y3 + ... + xny1) - (y1x2 + y2x3 + ... + ynx1)|

Using this formula with the given points, we get:

area = 1/2 * |(-6 * (-4) + (-1) * 1 + 3 * 0) - (0 * (-1) + (-4) * 3 + 1 * (-6))| = 1/2 * |(24 + (-1) + 0) - (0 + (-12) + (-6))| = 1/2 * |23| = 11.5

Therefore, the area of the figure determined by the given points is 11.5 square units.

To find the perimeter of the figure, we need to find the lengths of the three sides and add them up. Using the distance formula, we get:

1. The distance between (-6, 0) and (-1, -4) is:
2. sqrt[(-1 - (-6))^2 + (-4 - 0)^2] = sqrt[5^2 + (-4)^2] = sqrt[41]
3. The distance between (-1, -4) and (3, 1) is:
4. sqrt[(3 - (-1))^2 + (1 - (-4))^2] = sqrt[4^2 + 5^2] = sqrt[41]
5. The distance between (3, 1) and (-6, 0) is:
6. sqrt[(-6 - 3)^2 + (0 - 1)^2] = sqrt[(-9)^2 + (-1)^2] = sqrt[82]

Therefore, the perimeter of the figure determined by the given points is:

sqrt[41] + sqrt[41] + sqrt[82] ≈ 20.07 units.

Answer 2

now it is easy

distance formula has two sides at √41 unit each and other side at √82 unit

sure area can be found with Heron's formula for triangle area

(for fun see area of cyclic quadrilateral knowing four sides and proof)

or simply area=41/2 unit²,,,,because by observation triangle is right triangle

so area=(1/2)(legside1)(legside2)

If just given sides a,b,c and triangle is not right triangle

can use Heron's area formula, area=√[s(s-a)(s-b)(s-c)] ,,,,,,,,s=(a+b+c)/2

or determine angle A with a²=b²+c²-2bc*cos(A)

then area=(1/2)bc*sin(A)