2.9 Properties of Quadrilaterals.notebook
March 04, 2015
2.9 Properes of Quadrilaterals ‐ day 1
Reminder of Definions of Quadrilaterals:
Shape
Parallelogram
Rectangle
Square
Diagram
Definion
Both pairs of opposite sides are parallel.
A parallelogram with 4 right angles.
A rectangle with all sides the same length.
Rhombus
A parallelogram with all sides the same length.
Trapezoid
One pair of parallel sides.
Kite
Pairs of adjacent sides are equal.
2.9 Properties of Quadrilaterals.notebook
March 04, 2015
Investigate!
In groups of 4:
• Determine properes of the diagonals of all 6 quadrilaterals: Each person must draw 3 different quadrilaterals on grid paper. Each group should have 2 of each type of quadrilateral. Create the diagonals with spaghe. Use your diagram and the spaghe to answer the following quesons.
‐ Are the diagonals equal in length?
‐ Do they bisect each other?
‐ Do they intersect at a right angle?
• Draw the midsegments of a quadrilateral. What shape do you get?
• Draw and cut out a trapezoid. Fold it in half so that the parallel sides line up. What do you noce about the fold line? How does its length relate to the lengths of the parallel sides? How could you draw the fold line without folding?
2.9 Properties of Quadrilaterals.notebook
The diagonals of a parallelogram bisect each other.
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March 04, 2015
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2. Joining the midpoints of adjacent sides of any quadrilateral forms a parallelogram.
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3. The line segment joining the midpoints of the non‐parallel sides of a trapezoid is parallel to the parallel sides. Its length is equal to the mean of the lengths of the parallel sides.
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a
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2.9 Properties of Quadrilaterals.notebook
March 04, 2015
Conclusions for Diagonal Properes:
Shape
Equal
Lengths?
Perpendicular?
Bisect Each
Other?
Square
Yes
Yes
Yes
Rectangle
Yes
No
Yes
Parallelogram
No
No
Yes
Rhombus
No
Yes
Yes
Trapezoid
No
No
No
Kite
No
Yes
One does
2.9 Properties of Quadrilaterals.notebook
March 04, 2015
Using your formulas for slope, midpoint, and distance,what would you have to do to prove that a quadrilateral is a.....
‐slopes of opposite sides must be equal
‐ find lengths of all 4 sides
‐ pairs of adjacent sides must be equal
kite
‐ use distance formula to find all 4 lengths...they must all be equal
trapezoid
‐ slopes of one pair of opposite sides must be equal
‐ isosceles trap: 2 non‐parallel sides have equal length
‐ use distance formula to find all 4 lengths...they must all be equal
‐ slopes of two pairs of opposite sides must be equal
parallelogram
‐ use slope formula to show that we have 2 pairs of equal slopes that are negave reciprocals
rhombus
square
‐ lengths of all 4 sides must be equal
‐ use distance formula to find all 4 lengths...opposite sides must be equal
‐ lengths of all 4 sides must be equal
‐ use slope formula to show that we have 2 pairs of equal slopes that are ‐ opposite sides must have equal slopes
negave reciprocals
‐ adjacent sides must have negave reciprocal slopes
rectangle
‐ lengths of opposite sides must be equal
‐ opposite sides must have equal slopes
‐ adjacent sides must have negave reciprocal slopes
2.9 Properties of Quadrilaterals.notebook
Homework:
Page 134 #1,2,3,8,10
March 04, 2015
2.9 Properties of Quadrilaterals.notebook
March 04, 2015
Page 126 #15
Given D(‐18,12), E(‐6,‐12) and F(12,6), show that the right bisectors of the sides of ΔDEF all intersect at point (‐4,4).
1. Find equaons of 3 perpendicular bisectors
‐ find slope of side
‐ take negave reciprocal of slope
‐ find midpoint
‐ find equaon using midpoint and slope
2. Solve systems of equaons (substuon or eliminaon)
OR
Substute (‐4,4) in each equaon (LS = ... RS = ...) and verify that it works in all 3.
Attachments
2.9 diagonals of parallelogram.gsp
2.9 Varignon Parallelogram.gsp
2.9 Midsegment of Trapezoid.gsp