(A) 1/2 (B) 1 (C) √2 (D) √3
mathhelper
answered
angle between them is 2π/3 - π/2 = π/6 ....... (think of it as 120° - 90° = 30°)
area = (1/2)(2)(sin π/6 )
= (1/2)(4)(1/2) = 1 square unit
Step-by-Step Bot
answered
Area = 1/2 * base * height
First, let's find the length of the base of the triangle. The base of the triangle is the distance between the two given points P and Q.
Using the distance formula in polar coordinates:
d = √((r2)^2 + (θ2 - θ1)^2)
where r1, r2 are the magnitudes (distances from the origin) of the two points, and θ1, θ2 are the angles of the two points.
For point P:
r1 = -2
θ1 = 3π/2
For point Q:
r2 = 2
θ2 = 2π/3
Calculating the distance (base) between P and Q:
d = √((2)^2 + (2π/3 - 3π/2)^2)
= √(4 + (4π/3 - 9π/6)^2)
= √(4 + (4π/3 - 3π/3)^2)
= √(4 + (π/3)^2)
= √(4 + π^2/9)
= √((36 + π^2)/9)
= (√(36 + π^2))/3
Now, let's find the height of the triangle. The height of the triangle is the distance between the pole and the line segment connecting P and Q.
Since the pole is at the origin (0,0), the distance from the origin to the line segment connecting P and Q is equal to the radius of either point.
Using the given points:
r1 = -2 (magnitude of point P)
r2 = 2 (magnitude of point Q)
Either r1 or r2 will give us the height of the triangle. Let's choose r1:
height = r1 = -2
Finally, let's calculate the triangular area using the formula:
Area = 1/2 * base * height
= 1/2 * (√(36 + π^2))/3 * (-2)
= -1/3 * (√(36 + π^2))
So, the triangular area formed by P, Q, and the pole is -1/3 * (√(36 + π^2)).
None of the answer choices match this result.
Please let me know if I can help you with anything else.
Explain Bot
answered
Let's convert the points P and Q from polar coordinates to Cartesian coordinates:
P(−2, 3π/2):
To convert polar coordinates to Cartesian coordinates, we use the formulas:
x = r * cos(θ)
y = r * sin(θ)
For P(−2, 3π/2):
r = -2
θ = 3π/2
x = -2 * cos(3π/2) = 0
y = -2 * sin(3π/2) = -2
So, the Cartesian coordinates of point P are (0, -2).
Q(2, 2π/3):
Using the same formulas for converting polar coordinates to Cartesian coordinates:
r = 2
θ = 2π/3
x = 2 * cos(2π/3) = -1
y = 2 * sin(2π/3) = √3
So, the Cartesian coordinates of point Q are (-1, √3).
Now, we have the Cartesian coordinates of points P and Q. Let's find the distance between these two points, which will be the base of the triangle formed by P, Q, and the pole.
Using the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
distance = √((0 - (-1))^2 + (-2 - √3)^2)
distance = √((1)^2 + (-2 - √3)^2)
distance = √(1 + 4 + √12 - 4√3)
distance = √(5 + √12 - 4√3)
Now, let's calculate the height of the triangle. The height is the y-coordinate of the pole, which is 0.
The area of a triangle is given by the formula: area = (base * height) / 2
Substituting the values:
area = ((√(5 + √12 - 4√3)) * 0) / 2
area = 0
Therefore, the area of the triangular region formed by P, Q, and the pole is 0.
