Transcript 8.5 Rhombi and Squares

8.5 Rhombi and Squares
Objectives

Recognize and apply properties of
rhombi

Recognize and apply properties of
squares
Rhombi

A rhombus is a parallelogram with four
congruent sides.
Rhombi

Since rhombi are parallelograms, they have
all the properties of a parallelogram.

In addition, they have 2 other properties
which are theorems:
- the diagonals of a rhombus are ┴
- each diagonal of a rhombus bisects a
pair of opposite s
Example 1:
Given: BCDE is a rhombus,
and
Prove:
D
Example 1:
Proof: Because opposite angles of a rhombus are
congruent and the diagonals of a rhombus bisect each
other,
By substitution,
by the Reflexive Property and it is given that
Therefore,
by SAS.
Your Turn:
Given: ACDF is a rhombus;
Prove:
Your Turn:
Proof: Since ACDF is a rhombus, diagonals
bisect each other and are perpendicular to each other.
Therefore,
are both right
angles. By definition of right angles,
which means that
by
definition of congruent angles. It is given that
so
since alternate interior angles are
congruent when parallel lines are cut by a transversal.
by ASA.
Example 2a:
Use rhombus LMNP to find the value of y if
N
Example 2a:
The diagonals of a rhombus are
perpendicular.
Substitution
Add 54 to each side.
Take the square root of each side.
Answer: The value of y can be 12 or –12.
Example 2b:
Use rhombus LMNP to find
if
N
Example 2b:
Opposite angles are congruent.
Substitution
The diagonals of a rhombus bisect the angles.
Answer:
Your Turn:
Use rhombus ABCD and
the given information to
find the value of each
variable.
a.
Answer: 8 or –8
b.
Answer:
Squares

A square is a parallelogram with four
congruent sides and four right angles.
Squares

If a quadrilateral is both a rhombus and a
rectangle, then it is a square.
quadrilaterals
parallelograms
rhombi
rectangles
squares
Example 3:
Determine whether parallelogram ABCD is a rhombus,
a rectangle, or a square for A(–2, –1), B(–1, 3), C(3, 2),
and D(2, –2). List all that apply. Explain.
Explore Plot the vertices on a coordinate plane.
Example 3:
Plan
If the diagonals are perpendicular, then ABCD
is either a rhombus or a square. The diagonals
of a rectangle are congruent. If the diagonals are
congruent and perpendicular, then ABCD is a
square.
Solve Use the Distance Formula to compare the lengths
of the diagonals.
Example 3:
Use slope to determine whether the diagonals are
perpendicular.
Since the slope of
is the negative reciprocal of the
slope of
the diagonals are perpendicular.The lengths
of
and
are the same so the diagonals
are congruent. ABCD is a rhombus, a rectangle, and
a square.
Example 3:
Examine The diagonals are congruent and perpendicular
so ABCD must be a square. You can verify that
ABCD is a rhombus by finding AB, BC, CD, AD.
Then see if two consecutive segments are
perpendicular.
Answer: ABCD is a rhombus, a rectangle, and a square.
Your Turn:
Determine whether parallelogram EFGH is a rhombus,
a rectangle, or a square for E(0, –2), F(–3, 0), G(–1, 3),
and H(2, 1). List all that apply. Explain.
Your Turn:
Answer:
and
slope of
slope of
Since the slope of
is the
negative reciprocal of the slope of
, the
diagonals are perpendicular. The lengths of
and
are the same.
Example 4:
A square table has four legs
that are 2 feet apart. The table
is placed over an umbrella
stand so that the hole in the
center of the table lines up
with the hole in the stand.
How far away from a leg is
the center of the hole?
Let ABCD be the square formed by the legs of the table.
Since a square is a parallelogram, the diagonals bisect
each other. Since the umbrella stand is placed so that its
hole lines up with the hole in the table, the center of the
umbrella pole is at point E, the point where the diagonals
intersect. Use the Pythagorean Theorem to
find the length of a diagonal.
Example 4:
The distance from the center of the pole to a leg is equal
to the length of
Example 4:
Answer: The center of the pole is about 1.4 feet from a leg
of a table.
Your Turn:
Kayla has a garden whose length and width are each
25 feet. If she places a fountain exactly in the center of
the garden, how far is the center of the fountain from
one of the corners of the garden?
Answer: about 17.7 feet
Assignment

Pre-AP Geometry
Pg. 434 #12 – 23, 26 – 31, 40 – 42

Geometry:
Pg. 434 #12 – 23, 26 - 31