Question

Find the area of the triangle with the coorinates a) `(acostheta_1,bsintheta_1 , (acostheta_2,bsintheta_2 (acostheta_3,bsintheta_3` B) `(am_1^2, 2am_1) , (am_2^2, 2am_2), (am_3^2, 2am_3)`?

There are two separate sets of coordinates.

Answer 1

See below.

Explanation:

Given a triangle with vertices at

`p_1 = (x_1,y_1)`
`p_2=(x_2,y_2)`
`p_3 = (x_3,y_3)`

Area can be computed as

`A = 1/2 abs(det((x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)))`

so in the first case

`A = 1/2 abs(det((a m_1^2, a m_1, 1),(a m_2^2, a m_2, 1),(a m_3^2, a m_3, 1))) = a^2/2 abs((m_1-m_2)(m_1-m_3)(m_2-m_3))`

analogously

`A = (ab)/2abs(Sin(theta_1 - theta_2) + Sin(theta_3 - theta_1) + Sin(theta_2 - theta_3))`