Download 4.3

Math 53 "Winter ’09"
4.3 "The Rectangle, Square, and Rhombus"
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Objectives:
*
Understand the properties of the special parallelograms
Rectangle
Square
Rhombus
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Key Concepts:
Rectangle
Square
Rhombus
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The Rectangle
In this section, we investigate special parallelograms. Also, The Pythagorean Theorem, which deals with right triangles,
is useful in applications involving quadrilaterals that have right angles.
De…nition:
"Rectangle (rect.)"
A rectangle is a parallelogram that has a right angle
Corollary 4.3.1:
kAll angles of a rectangle are right anglesk
Theorem 4.3.2:
kThe diagonals of a rectangle are congruentk
Page: 1
Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
4.3
Example 1: (Proof of Theorem 4.3.2)
Given:
Rectangle M N P Q with diagonals M P and N Q
Prove:
MP = NQ
PROOF
Statements
Reasons
Example 2:
Given:
Prove:
6
Rectangle W XY Z with diagonals W Y and XZ
1=6 2
PROOF
Statements
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Reasons
Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
4.3
Example 3: (Using properties of rectangles)
Use the properties of rectangles to solve the following:
Given:
AB = x + y;
Find:
x and y:
BC = x + 2y;
CD = 2x
y
1; and DA = 3x
3y + 1
The Square
De…nition:
"Square"
A square is a rectangle that has two congruent adjacent sides
Corollary 4.3.3:
kAll sides of a square are congruentk
Example 4: (Using properties of squares)
On a softball diamond (actually a square), the distance along the base paths is 60 ft.
Find the distance from home plate to second base.
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Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
4.3
The Rhombus
De…nition:
"Rhombus"
A rhombus is a parallelogram with two congruent adjacent sides
Corollary 4.3.4:
kAll sides of a rhombus are congruentk
Theorem 4.3.5:
kThe diagonals of a rhombus are perpendiculark
Example 5: (Picture proof of theorem 4.3.5)
Study the picture proof of Theorem 4.3.5. In the proof, pairs of triangles are congruent by the reason SSS:
Example 6: (Using properties of rhombi)
Consider rhombus ABCD with diagonals AC and DB.
If AC = 14 and DB = 10 …nd BC:
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Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
4.3
When all vertices of a quadrilateral lie on a circle, the quadrilateral is a cyclic quadrilateral.
As it happens, all rectangles are cyclic quadrilaterals but no rhombus is a cyclic quadrilateral.
Example 7:
Find the perimeter of the cyclic quadrilateral shown.
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Bibiana Lopez