Q. 174.2( 13 Votes )

# 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.

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(x_{1},y_{1}) and B(x_{2},y_{2}) 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.

Rate this question :

By using the concept of slope, show that the points (– 2, – 1), (4, 0), (3, 3) and (– 3, 2) vertices of a parallelogram.

RD Sharma - MathematicsFind the angle between X - axis and the line joining the points (3, – 1) and (4, – 2).

RD Sharma - MathematicsThe line through the points (– 2, 6) and 94, 8) is perpendicular to the line through the points (8, 12) and (x, 24). Find the value of x.

RD Sharma - MathematicsWithout using the distance formula, show that points (– 2, – 1), (4, 0), (3, 3) and (– 3, 2) are the vertices of a parallelogram.

RD Sharma - MathematicsThe equation of the line through the intersection of the lines 2x – 3y = 0 and 4x – 5y = 2 and

The value of the λ, if the lines (2x + 3y + 4) + λ (6x – y + 12) = 0 are

Equation of the line passing through (1, 2) and parallel to the line y = 3x – 1 is

Mathematics - ExemplarThe equation of the straight line passing through the point (3, 2) and perpendicular to the line y = x is

Mathematics - Exemplar