Verifying Properties of Quadrilaterals

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1 Verifing roperties of uadrilaterals We can use the tools we have developed to find, classif, or verif properties of various shapes made b plotting coordinates on a Cartesian plane. Depending on the problem, we use three primar tools and a few ke facts. Need to Know: Slope Distance Midpoint Ke Facts: arallel lines have the same slope erpendicular lines have slopes that are negative reciprocals: roperties of uadrilaterals TO OVE Calculate the slopes of the sides and show that the slopes of opposite sides are equal and therefore the sides are bisect each other b calculating the midpoints and show that the are the same. Calculate the slopes of the sides and show that the slopes of opposite sides are equal and therefore the sides are Demonstrate that the slopes of adjacent sides are negative reciprocals of each other to prove that the meet at a 90 o (right) angle. bisect each other b showing that the meet at the same midpoint. The are also equal in length.

2 roperties of uadrilaterals Continued... TO OVE Calculate the slopes of all sides and show that the slopes of opposite sides are equal and therefore the sides are Demonstrate that the slopes of adjacent sides are negative reciprocals of each other and thus show that there is a 90 o (right) angle. Calculate the length of all sides and show that the are equal. Diagonals bisect each other, are equal in length and the diagonals are perpendicular to each other. A rhombus is sometimes referred to as a diamond. Calculate the slopes of all sides and show that the slopes of opposite sides are equal and therefore the sides are Calculate the length of all sides and show that the are equal. Diagonals bisect each other, are equal in length and the diagonals are perpendicular to each other. Note: Squares, ectangles and hombus are all arallelograms roperties of uadrilaterals Continued... It is called an Isosceles trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown in the diagram above. TO OVE Calculate the slopes of the sides and show that the slopes of two of the opposite sides are equal and therefore parallel and the slopes of the other two sides are not equal and therefore not Calculate the length of the sides and show that the length of two pairs of adjacent sides are equal. are perpendicular b finding their slopes and showing that the are negative reciprocals of each other.

3 Eample # : A surveor has marked the corners of a lot where a building is going to be constructed. The corners have coordinates of (-5,-5), (-0,0), (-5,5) and S (0,0). Use analtic geometr to identif the shape of the quadrilateral S S I see that the shape of the building lot looks like a parallelogram or a rhombus. I know that if S is either of these shapes, opposite sides would be If S is a rhombus, all the sides would have to be the same length. The slopes of and S are the same, so the are The slopes of and S are also the same, so the are parallel too. The length of all four sides are equal. I can therefore conclude that the building lot is a rhombus.

4 Eample # : A triangle has vertices at A (-,-), B (,0) and C(,). Classif the triangle. Justif our decision. C B 0 A After plotting the points on the grid paper, I think that the triangle might be isosceles since AB and BC look like the are the same length. The triangle might also be a right triangle since angle B looks like it might be 90 o. Since AB and BC are the same length, the triangle is isosceles. The slopes of AB and BC are negative reciprocals, so AB is perpendicular to BC. This means that the triangle is a right triangle. Eample # : Show that the midsegments of the quadrilateral, with vertices at (-7,9), (9,), (9,-) and S(,-) form a parallelogram. Midsegments are a line segment that connects the midpoints of two adjacent sides in a quadrilateral. J H L S M K I used the midpoint formula to determine the coordinates of the midpoints of,, S and S, which are labelled as J, K, L and H. I needed to show that JK is parallel to LH and that KL is parallel to HJ. I used the slope formula to calculate the slopes of JK, KL, LH and HJ. The slopes JK and LH are the same and the slopes of KL and HJ are the same. This means opposite sides of the quadrilateral are parallel, therefore JKLH is a parallelogram.

5 Eample # : A triangle has vertices at (-,), (,) and (,-). Show that the midsegment joining the midpoints of and is parallel to and half its length. 5 J L Homework: p. 8 #, 5 0, bc, ace