Midpoints and Diagonals in Quadrilaterals
Warm up:
1. Idenfy the value of the missing side. Support your answer
Sol: DE=BC2
BC=2DE
x=218
∴x=36 units
2. The area of ∆ABD is 24 cm2. Determine the areas of ∆ACD and ∆ABC. Support your answers.
Sol: AreaACD=AreaABD
=24mc2
AreaABC=2AreaABD
=224
=48cm2
Example 1: a) What type of quadrilateral is ABCD?
Quadrilateral ABCD is a trapezoid
b) How can the measurements be used to prove what type of quadrilateral ABCD is?
Side length are different.
Slopes of opposite side are different.
Example 2: a) What type of quadrilateral is ABCD? Draw the diagonals
Draw the diagonals
Record their lengths
AC=6.05 units
BD=7.56 units
Do the diagonals bisect each other?
Yes, they do.
b) Are the diagonals perpendicular? How could you check this?
I: Check whether the product of the slope AC and BD is equal to-1
II: Measure one of the angles where the diagonals intersect.
∴ The diagonals are not perpendicular.
Example 3: Name each of the following quadrilaterals
For which quadrilaterals do the diagonals bisect each other?
-Square, rectangle, parallelogram and rhombus
For which quadrilaterals are the diagonals perpendicular to each other?
-Square and rhombus
For which quadrilaterals are the diagonals perpendicular bisectors of each other?
-Square, kite and rhombus
What do the diagonals of a rhombus do to the angles at the vertices?
The diagonals split the angles in half
Conclusions:
• Joining the midpoints of the sides of any quadrilateral produces a parallelogram.
• The diagonals of a parallelogram bisect each other.
• The diagonals of a square are equal in length and bisect each other at right angles.
• The diagonals of a rectangle are equal and bisect each other.
• The diagonals of a kite are perpendicular to each other.
• The diagonals of a rhombus bisect each other at right angles and bisect the angles at the vertices.
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