The vertices of a quadrilateral are A(-4, -2), B(2, 6), C(8, 5) and D(9, -7). Using slopes, show that the midpoints of the sides of the quad. ABCD from a parallelogram.

Question

Philoid · Accepted Answer

The vertices of the given quadrilateral are A(-4,-2) B(2, 6), C(8, 5) and D(9, -7)

The mid point of a line A(x₁,y₁) and B(x₂,y₂) is found out by

Now midpoint of AB =

The midpoint of BC =

The midpoint of CD =

Midpoint of DA =

So now we have four points

P(-1,2),Q(5,5.5),R(8.5,-1),S(2.5,-4.5)

Slope of PQ =

Slope of QR =

Slope of RS =

Slope of SP =

Now we can observe that slope of PQ = RS and slope of QR = SP

Which shows that line PQ is parallel to RS and line QR is parallel to SP

Also, the product of two adjacent lines is not equal to -1

Therefore PQRS is a parallelogram.