Question

Prove that the points (– 4, – 1), (– 2, – 4), (4, 0) and (2, 3) are the vertices of a rectangle.

Answer 1

To Prove: Given vertices are of the rectangle. 

Explanation: We have given points A(– 4, – 1), B(– 2, – 4), C(4, 0) and D(2, 3) 

The points are joining in the form of AB, BC, CD, and AD 

The formula used: The slope of the line, m = \(\frac{y_2-y_1}{x_2-x_1}\)

Now, The slope of Line AB, m_AB = \(\frac{-4-(-1)}{-2-(-4)}\) 

m_AB = \(\frac{-3}{2}\)

The slope of BC, m_BC = \(\frac{0-(-4)}{4-(-2)}\)

m_BC = \(\frac{4}{6}=\frac{3}{2}\) 

Now, The slope of Line CD, m_CD = \(\frac{3-0}{2-4}\) 

m_CD = \(\frac{3}{-2}\)

The slope of AD, m_AD =  \(\frac{3-(-1)}{2-(-4)}\)

m_AD = \(\frac{4}{6}=\frac{2}{3}\)

Here, We can see that, m_AB = m_CD and m_BC = m_AD

i.e, AB CD and BC AD 

And, m_AB x m_BC = \(-\frac{3}{2}\times\frac{2}{3}\) = -1

M_CD x m_AD = \(-\frac{3}{2}\times\frac{2}{3}\) =-1

So, that AB||BC and CD||AD 

Hence, ABCD is a Rectangle.