Question

find the area of the triangle formed by P(-2, 2pi/) Q(2, 4pi/3) and the pole

Answer 1

AAAaannndd the bot gets it wrong yet again!
all that work to get a wrong answer.
A = 1/2 * 2 * 2 * sin π/3 = √3

Answer 2

The area of a triangle is given by the formula A = 1/2 * base * height.

The base of the triangle is the distance between points P and Q, which can be calculated using the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

d = sqrt((2 - (-2))^2 + (4pi/3 - 2pi/3)^2)

d = sqrt(4^2 + (2pi/3)^2)

d = sqrt(16 + (4pi^2)/9)

d = sqrt(16 + (36pi^2)/9)

d = sqrt(16 + 36pi^2)/3

The height of the triangle is the distance from the pole to the line segment connecting P and Q. This can be calculated using the distance formula:

h = sqrt((x - x1)^2 + (y - y1)^2)

h = sqrt((0 - (-2))^2 + (0 - 2pi/3)^2)

h = sqrt(2^2 + (2pi/3)^2)

h = sqrt(4 + (4pi^2)/9)

h = sqrt(4 + (36pi^2)/9)

h = sqrt(4 + 36pi^2)/3

The area of the triangle is then:

A = 1/2 * d * h

A = 1/2 * (sqrt(16 + 36pi^2)/3) * (sqrt(4 + 36pi^2)/3)

A = (sqrt(16 + 36pi^2) * sqrt(4 + 36pi^2))/9

A = (sqrt(20 + 72pi^2))/9