Download - LESSON TWENTY-THREE: RHOMBI APOCALYPSE!!!
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LESSON TWENTY-THREE:
RHOMBI APOCALYPSE!!!
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RHOMBI AND SQUARES
• So right now we have discussed the definition and properties of rectangles and parallelograms.
• Two new figures we will be discussing today are rhombi and squares.
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RHOMBI AND SQUARES
• A rhombus is a parallelogram with four congruent sides.
• Since a rhombus is a parallelogram, it has all the properties that a parallelogram does.
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RHOMBI AND SQUARES• Those properties are…– Opposite sides are parallel and congruent.– Opposite angles are congruent.– Consecutive angles are supplementary.– The diagonals bisect each other.
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RHOMBI AND SQUARES
• Rhombi have properties also that are all their own.
• The first says that if a parallelogram is a rhombus, his its diagonals are perpendicular.
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RHOMBI AND SQUARES
• The second property states that if a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.
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RHOMBI AND SQUARES
• The other figure, we will discuss today is called a square.
• A square is the most specific of the figures because we can think of it as a combination of a rhombus and rectangle.
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RHOMBI AND SQUARES
• A square is a parallelogram with four right angles and four congruent sides.
• Because a square is both a rhombus and rectangle, the special properties for rhombi and rectangles, both apply to squares.
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RHOMBI AND SQUARES
• Therefore if a parallelogram is a square, then…– Its diagonals are congruent.– Its diagonals are perpendicular.– Its diagonals bisect a pair of opposite angles.
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RHOMBI AND SQUARES
• Using what we know about the distance formula, we can now identify what figure is represented on a plane.
• What type of figure would have 4 congruent sides and 4 congruent diagonals?
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RHOMBI AND SQUARES• For the figure below our points are (-3, 0), (-1, 3),
(4, 3), (2, 0).• Is this a rhombus, rectangle, square or none of
these?
A
D C
B
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RHOMBI AND SQUARES• In solving these distances we find that AB • This equals = = • By this process, we find that DC is also• We find that AD and BC equal
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RHOMBI AND SQUARES• Finally, let’s find the diagonals.• Using the distance formula, we find that AC =
and BD = • So we have opposite sides that are congruent,
but non-congruent diagonals.• Is this a rhombus, rectangle, square or none of
these?
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RHOMBI AND SQUARES
• What type of figure would have all four sides congruent, but non-congruent diagonals?
• This is the type of problem you’ll be practicing today and the logic you’ll need for it.