8.4 properties of rhombuses, rectangles, and squares€¦ · squares a square is a regular...
TRANSCRIPT
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
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
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.
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.
Example 3
For any rhombus QRST, decide whether the
statement is always or sometimes true.
Draw a sketch and explain your reasoning.
1. Q S
2. Q R
Example 4
For any rectangle ABCD, decide whether the
statement is always or sometimes true.
Draw a sketch and explain your reasoning.
1. AB CD
2. AB BC
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.
Example 6
The previous theorem is a biconditional.
Write the two conditional statements that
must be proved separately to prove the
entire theorem.
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?
Diagonal Theorem 3
A parallelogram is a rhombus if and only if
each diagonal bisects a pair of opposite
angles.
Example 8
Prove that if a parallelogram has
perpendicular diagonals, then it is a
rhombus.
Given: ABCD is a
parallelogram;
AC BD
Prove: ABCD is a
rhombus
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?
Rhombus Area
Since a rhombus is a
parallelogram, we
could find its area
by multiplying the
base and the
height.A b h
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
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.
Exercise 12
If the length of each diagonal of a rhombus is
doubled, how is the area of the rhombus
affected?