assignment p. 537-540: 1, 2, 3- 48 m3, 49, 52, 55, pick one (56, 60, 61, 63) p. 723: 5, 18, 25, 27,...
TRANSCRIPT
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Assignment
• P. 537-540: 1, 2, 3-48 M3, 49, 52, 55, Pick one (56, 60, 61, 63)
• P. 723: 5, 18, 25, 27, 40
• P. 732: 8, 11, 15, 20, 28, 36
• Challenge Problems
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Rhombuses Or Rhombi
What makes a quadrilateral a rhombus?
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Rhombuses Or Rhombi
A rhombus is an equilateral parallelogram.– All sides are
congruent
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Rhombus Corollary
A quadrilateral is a rhombus if and only if it has four congruent sides.
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Rectangles
What makes a quadrilateral a rectangle?
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Rectangles
A rectangle is an equiangular parallelogram.
• All angles are congruent
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Example 1
What must each angle of a rectangle measure?
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Rectangle Corollary
A quadrilateral is a rectangle if and only if it has four right angles.
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Squares
What makes a quadrilateral a square?
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Squares
A square is a regular parallelogram.
• All angles are congruent
• All sides are congruent
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Square Corollary
A quadrilateral is a square if and only if it is a rhombus and a rectangle.
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8.4 Properties of Rhombuses, Rectangles, and Squares
Objectives:
1. To discover and use properties of rhombuses, rectangles, and squares
2. To find the area of rhombuses, rectangles, and squares
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Example 2
Below is a concept map showing the relationships between some members of the parallelogram family. This type of concept map is known as a Venn Diagram. Fill in the missing names.
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Example 2
Below is a concept map showing the relationships between some members of the parallelogram family. This type of concept map is known as a Venn Diagram.
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Example 3
For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.
1. Q S2. Q R
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Example 4
For any rectangle ABCD, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.
1. AB CD2. AB BC
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Example 5
Classify the special quadrilateral. Explain your reasoning.
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Investigation 1
We know that the diagonals of parallelograms bisect each other. The diagonal of rectangles and rhombuses have a few other properties we will discover using GSP.
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Diagonal Theorem 1
A parallelogram is a rectangle if and only if its diagonals are congruent.
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Example 6
The previous theorem is a biconditional. Write the two conditional statements that must be proved separately to prove the entire theorem.
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Example 7
You’ve just had a new door installed, but it doesn’t seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?
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Diagonal Theorem 2
A parallelogram is a rhombus if and only if its diagonals are perpendicular.
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Diagonal Theorem 3
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
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Example 8
Prove that if a parallelogram has perpendicular diagonals, then it is a rhombus.
Given: ABCD is a
parallelogram;
AC BDProve: ABCD is a
rhombus
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Example 9: SAT
In the figure, a small square is inside a larger square. What is the area, in terms of x, of the shaded region?
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Example 10
In the diagram below, MRVU SPTV. Let the area of MRVU equal A. Show that A = bh.
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Rhombus Area
Since a rhombus is a parallelogram, we could find its area by multiplying the base and the height.
A b h
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Rhombus Area
However, you’re not always given the base and height, so let’s look at the two diagonals. Notice that d1 divides the rhombus into 2 congruent triangles.
Ah, there’s a couple of triangles in there.
1
2A b h
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Rhombus Area
So find the area of one triangle, and then double the result.
1
2A b h
12
2A b h
1 2
1 12
2 2A d d
1 2
12
4A d d
1 2
1
2d d
1 2
1
2A d d
Ah, there’s a couple of triangles in there.
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Polygon Area Formulas
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Exercise 11
Find the area of the shaded region.1. 2. 3.
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Exercise 12
If the length of each diagonal of a rhombus is doubled, how is the area of the rhombus affected?
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Assignment
• P. 537-540: 1, 2, 3-48 M3, 49, 52, 55, Pick one (56, 60, 61, 63)
• P. 723: 5, 18, 25, 27, 40
• P. 732: 8, 11, 15, 20, 28, 36
• Challenge Problems