PERIMETERS
A perimeter is the measure of the distance AROUND an object.
l
S
S
S
w
w
S
l
Perimeter of a Rectangle
=w+l+w+l
= 2w + 2l
Perimeter of a Square
= S +S+ S + S
= 4S
Triangles
Scalene Triangle
l2
l1
Isosceles Triangle
(2 sides and 2 angles are equal)
C
l3
Perimeter of a Scalene Triangle
= l 1 + l 2 + l3
Equilateral Triangle
(3 sides and 3 angles are equal)
s
C
A
s
B
s
s = AC = BC =AB
A =
B =
C
Perimeter of an Equilateral Triangle
=s+s+s
= 3s
s
s
B
A
t
S=AC=BC
A =
Perimeter of an
Isoceles Triangle
=s+s+t
= 2s + t
B
Example:
The perimeter of a rectangle is 26 ft. The length of the rectangle is 1 ft more than twice
the width. Find the width and length of the rectangle.
Step 1) What are we trying to find? The width and length of the rectangle.
Let w = width, and l = length.
Given info: Perimeter is 26 ft. It is a rectangle, so the formula for a rectangle’s perimeter
is P = 2w + 2l.
Also, length of the rectangle is 1 ft more than twice the width.
Step 2) Make an equation from given info.
Perimeter = 26 ft = 2w + 2l
Length is 1 ft more than twice its width.
l
= 1
+
2w
We can combine these equations to solve for each variable, w and l.
26 = 2w + 2l
Substitute equation, l = 1 + 2w for l in the above equation.
26 = 2w + 2(1 + 2w)
Step 3) Solve equation
Use distributive property to get rid of parentheses.
26 = 2w + 2 + 4w
Combine like terms
26 = 6w + 2
24 = 6w
4=w
What about l? l = 1 + 2w = 1 + 2(4) = 1+8=9
Step 4) Check result. Perimeter with w=4 and l=9 should be 26
26 = 2(4) + 2(9)= 8 + 18 = 26 Yes.
Step 5) State conclusion (Remember the measuring units!)
The width of the rectangle is 4 ft and the length is 9 ft.
Example 1 The perimeter of an isosceles triangle is 25 ft.
The length of the third side is 2 ft less than the length of one of
the equal sides. Find the measures of the three sides of the
triangle.
Now you try this one:
A carpenter is designing a square patio with a perimeter of 52 ft.
What is the length of each side?
Angles are formed by two rays, lines, or line segments with a common
endpoint. The common endpoint is called the vertex. In the angle
below, point A is the vertex. The angle can be denoted as
BAC or
CAB
Note that the vertex, point A, must be in the middle.
B
vertex
The unit of measurement of an angle is degrees.
A
C
Right angle
90°
Acute angle
Less than 90°
Straight angle
Obtuse angle
180°
Greater than 90°
B
C
Intersection Point
A
E
D
Intersecting lines
form angles at their
intersection point.
Angles that share a
common side are
called adjacent
angles. The angles
that are not
adjacent are called
vertical angles.
Example
EAB and BAC are adjacent angles.
BAC and CAD are adjacent angles.
EAD and BAC are vertical angles.
(3x+15)°
(4x-20)°
The vertical angles property allows us to make an equation:
3x+15 = 4x -20 and solve for x.
Vertical angles have equal measures of degrees.
We call this being “congruent.”
Two angles are supplementary angles when the sum of their measures is 180°.
When two lines intersect, adjacent angles are supplementary because the sides that
are not in common form a straight angle.
EAD and
DAC are supplementary angles.
DAC = 180°
Two angles are complementary angles
when the sum of their measures is 90°.
When two adjacent angles form a right
angle with the sides that are not in common,
these angles are complementary because the
measures add up to 90°.
B
C
90°
A
EAD +
D
BAC and
CAD are complementary angles.
Parallel lines are lines in the same plane that never intersect
(that have the same slope). If two lines, l1 and l2 are parallel,
we say l1 || l2.
Transversal – a line that intersects two or more lines on
the same plane.
A
Alternate Interior Angle Property
A transversal intersects parallel lines at
congruent angles.
Because of this and also because of the property
of supplementary angles and the property of
vertical angles, we can rewrite the diagram on
the left as this:
B
C
D
E
F
G
H
A
The angles on the inside of the parallel
lines that are congruent are called
“Alternate Interior Angles.”
C=
C and
F
D=
D and
E
F are alternate interior angles.
E are alternate interior angles.
180° - A
180° - A
A
A
180° - A
180° - A
A
Example
Given that l1|| l2, solve for x
l1
(3x+20)°
l2
(3x-80)°
(3x-80)°
l1
(3x+20)°
l2
These two angles are not equal, but
we can still solve for x by using
other properties.
Since these two lines are parallel,
the transveral intersects them at
congruent angles, so the angle
adjacent to (3x+20)° is (3x-80)°.
(3x-80)°
The supplementary angle property
says that if two adjacent angles
form a straight angle with
uncommon sides, they are
supplementary, so
(3x – 80) + (3x + 20) = 180.
Now that we have an equation, we
can solve for x.
The Sum of the Measures of the Interior Angles of a Triangle
is 180°
The sum of angles in a triangle is always 180 degrees. A
rigorous proof takes some work, but the statement can be made
plausible by the following argument. In the triangle ABC, draw a
line through point C that is parallel to AB. This creates two
additional angles, A' and B'.
The three angles (A',C,B') add up to 180 degrees, because they
are adjacent to each other and backed off against a straight line.
However, by the properties of parallel lines,
Angle A = Angle A' and
Angle B = Angle B'.
Therefore (A,C,B) also add up to 180 degrees.
l1
l1 || l2
l2
Example 5:
Given that Angle a = 45°
and Angle x = 100°, find the measure of angles b,c, and y.
x
b
a
k
Angle b =
Angle c =
Angle y =
y
c
l
Try this one:
Given that Angle y = 55 °, and that lines m and k are perpendicular.
find the measures of angles a, b, and d.
m
d
k
b
a
y
Angle a =
Angle b =
Angle d =
Chapter 4. 4 Markup and Discount
$ Retail Price = $Cost + $ Markup
Amount the store
raises the price to
make a profit
Selling price for an item
Amount the store paid for the item
Markup = Markup Rate * $Cost
$ Sale Price= $ Original Price – $ Discount
$Discount = Discount Rate * $Original Price
Markup rate must be converted from
percent to decimal before multiplying .
Original Price is the “Regular Price” of the
item.
Example
The manager of a clothing store buys a suit for $180 and sells that suit for $252. Find
the markup rate.
$Markup = Markup Rate * $Cost
So
Markup Rate = $Markup/ $Cost
We know the cost of the suit, that is $180.
What is the Markup?
We can use the other formula, $Retail Price = $Cost + $Markup
to find Markup.
$Markup = $Retail Price - $Cost
= $252 - $180
= $72
Markup Rate = $Markup/ $Cost = $72/$180 = 0.4
Converting 0.4 to percent, we get 40%.
Conclusion: Markup Rate is 40%
You try this: The cost to a sporting goods store of a tennis racket is $120. The selling
price of the racket is $180. What is the markup rate?