Proving Quadrilaterals on the Coordinate Plane

February 27, 2013

Prove that the quadrilateral ABCD with vertices A(-5, -1), B(-9,6), C(-1,5), and

D(3,-2) is a rhombus.

Do the work (Calculations)Find the Slopes of the sides.

Slope of CD=

Slope of AD=

Slope of AB=

Slope of BC=

Do the work (calculations Find the Lengths of the Sides

CD=

===

AD=

===

AB=

= ==

BC====

So Why is it a Rhombus?

• Cause it looks like one.• The sides are equal.• It’s a rhombus.

So Why is it a Rhombus?

Possible Answer: Sides AB and CD are parallel because they have the same slope (). Also, sides BC and AD are parallel because they have the same slope (). When opposite sides are parallel, the shape is a parallelogram. All fours sides of the shape are the same length (). A parallelogram with all equal sides is a rhombus so the shape is a rhombus.

2. Prove that ABCD with A(–5, 0), B(2, –4), C(6, 3), and D(–1, 7) is a rectangle.Get into your groups of 4 and discuss and solve the second problem.

Prove that ABCD with A(–5, 0), B(2, –4), C(6, 3), and D(–1, 7) is a rectangle.

Prove that ABCD with A(–5, 0), B(2, –4), C(6, 3), and D(–1, 7) is a rectangle.

Calculate all the Slopes.

Possible AnswerAB and CD have the same slope (), which means these sides are parallel. BC and AD have the same slope (), which means these sides are parallel. Since opposite sides are parallel, the shape is a parallelogram. AB is perpendicular to BC because they have opposite reciprocal slopes(). Also BC is perpendicular to CD and CD is perpendicular to AD and AD is perpendicular to AB because they also have the same opposite reciprocal slopes. Perpendicular lines form right angles, which means all four angles of the shape are right angles. This means that our parallelogram is also a rectangle.