1-5
1-5 Using
UsingFormulas
FormulasininGeometry
Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
Geometry
1-5 Using Formulas in Geometry
Warm Up
Evaluate. Round to the nearest hundredth.
1. 122 144
2. 7.62 57.76
3.
8
4.
7.35
5. 32() 28.27
6. (3)2 88.83
Holt Geometry
1-5 Using Formulas in Geometry
Objective
Apply formulas for perimeter, area, and
circumference.
Holt Geometry
1-5 Using Formulas in Geometry
Vocabulary
perimeter
area
base
height
Holt Geometry
diameter
radius
circumference
pi
1-5 Using Formulas in Geometry
The perimeter P of a plane figure is
the sum of the side lengths of the
figure.
The area A of a plane figure is the
number of non-overlapping square
units of a given size that exactly
cover the figure.
Holt Geometry
1-5 Using Formulas in Geometry
Holt Geometry
1-5 Using Formulas in Geometry
The base b can be any side of a
triangle. The height h is a segment
from a vertex that forms a right angle
with a line containing the base. The
height may be a side of the triangle or
in the interior or the exterior of the
triangle.
Holt Geometry
1-5 Using Formulas in Geometry
Remember!
Perimeter is expressed in linear
units, such as inches (in.) or meters
(m). Area is expressed in square
units, such as square centimeters
(cm2).
Holt Geometry
1-5 Using Formulas in Geometry
Example 1A: Finding Perimeter and Area
Find the perimeter and area of each figure.
= 2(6) + 2(4)
= 12 + 8 = 20 in.
= (6)(4) = 24 in2
Holt Geometry
1-5 Using Formulas in Geometry
Example 1B: Finding Perimeter and Area
Find the perimeter and area of each figure.
P=a+b+c
= (x + 4) + 6 + 5x
= 6x + 10
Holt Geometry
= 3x + 12
1-5 Using Formulas in Geometry
Check It Out! Example 1
Find the perimeter and area of a square with
s = 3.5 in.
P = 4s
A = s2
P = 4(3.5)
A = (3.5)2
P = 14 in.
A = 12.25 in2
Holt Geometry
1-5 Using Formulas in Geometry
Example 2: Crafts Application
The Queens Quilt block includes 12 blue
triangles. The base and height of each triangle
are about 4 in. Find the approximate amount of
fabric used to make the 12 triangles.
The area of one triangle is
The total area of the 12 triangles is
12(8) = 96 in2.
Holt Geometry
1-5 Using Formulas in Geometry
Check It Out! Example 2
Find the amount of fabric used to make four
rectangles. Each rectangle has a length of
and a width of
in.
The area of one rectangle is
The amount of fabric to make four rectangles is
Holt Geometry
in.
1-5 Using Formulas in Geometry
In a circle a diameter is a segment that
passes through the center of the circle and
whose endpoints are on a circle. A radius of
a circle is a segment whose endpoints are the
center of the circle and a point on the circle.
The circumference of a circle is the distance
around the circle.
Holt Geometry
1-5 Using Formulas in Geometry
The ratio of a circle’s circumference to its
diameter is the same for all circles. This ratio
is represented by the Greek letter  (pi). The
value of  is irrational. Pi is often
approximated as 3.14 or
.
Holt Geometry
1-5 Using Formulas in Geometry
Example 3: Finding the Circumference and
Area of a Circle
Find the circumference and area of a circle with
radius 8 cm. Use the  key on your calculator.
Then round the answer to the nearest tenth.
 50.3 cm
Holt Geometry
 201.1 cm2
1-5 Using Formulas in Geometry
Check It Out! Example 3
Find the circumference and area of a circle
with radius 14m.
 88.0 m
Holt Geometry
 615.8 m2
1-6
Midpoint and Distance
in the Coordinate Plane
Objectives
Develop and apply the formula for midpoint.
Use the Distance Formula and the
Pythagorean Theorem to find the distance
between two points.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Vocabulary
coordinate plane
leg
hypotenuse
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
A coordinate plane is a plane that is
divided into four regions by a horizontal
line (x-axis) and a vertical line (y-axis) .
The location, or coordinates, of a point are
given by an ordered pair (x, y).
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
You can find the midpoint of a segment by
using the coordinates of its endpoints.
Calculate the average of the x-coordinates
and the average of the y-coordinates of the
endpoints.
Holt Geometry
Ordered pairs are used to locate
points in a coordinate plane.
y-axis (vertical axis)
5
5
-5
x-axis (horizontal
axis)
-5
origin (0,0)
In an ordered pair, the first number is
the x-coordinate. The second number
is the y-coordinate.
Graph. (-3, 2)
5
•
5
-5
-5
What is the ordered pair for A?
1.
2.
3.
4.
(3, 1)
(1, 3)
(-3, 1)
(3, -1)
5
•A
5
-5
-5
What is the ordered pair for B?
1.
2.
3.
4.
(3, 2)
(-2, 3)
(-3, -2)
(3, -2)
5
5
-5
•B
-5
What is the ordered pair for C?
1.
2.
3.
4.
(0, -4)
(-4, 0)
(0, 4)
(4, 0)
5
5
-5
•
C
-5
What is the ordered pair for D?
1.
2.
3.
4.
(-1, -6)
(-6, -1)
(-6, 1)
(6, -1)
5
5
-5
•D
-5
Write the ordered pairs that name
points A, B, C, and D.
A = (1, 3)
B = (3, -2)
C = (0, -4)
D = (-6, -1)
5
•A
5
-5
•D
•B
•
C
-5
The x-axis and y-axis separate the
coordinate plane into four
regions, called quadrants.
II
(-, +)
III
(-, -)
I
(+, +)
IV
(+, -)
Name the quadrant in which each
point is located
(-5, 4)
1.
2.
3.
4.
5.
6.
I
II
III
IV
None – x-axis
None – y-axis
Name the quadrant in which each
point is located
(-2, -7)
1.
2.
3.
4.
5.
6.
I
II
III
IV
None – x-axis
None – y-axis
1-6
Midpoint and Distance
in the Coordinate Plane
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Helpful Hint
To make it easier to picture the problem, plot
the segment’s endpoints on a coordinate
plane.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 1: Finding the Coordinates of a Midpoint
Find the coordinates of the midpoint of PQ
with endpoints P(–8, 3) and Q(–2, 7).
= (–5, 5)
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 1
Find the coordinates of the midpoint of EF
with endpoints E(–2, 3) and F(5, –3).
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 2: Finding the Coordinates of an Endpoint
M is the midpoint of XY. X has coordinates
(2, 7) and M has coordinates (6, 1). Find
the coordinates of Y.
Step 1 Let the coordinates of Y equal (x, y).
Step 2 Use the Midpoint Formula:
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 2 Continued
Step 3 Find the x-coordinate.
Set the coordinates equal.
Multiply both sides by 2.
12 = 2 + x
– 2 –2
Simplify.
Subtract.
10 = x
Simplify.
The coordinates of Y are (10, –5).
Holt Geometry
2=7+y
– 7 –7
–5 = y
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 2
S is the midpoint of RT. R has coordinates
(–6, –1), and S has coordinates (–1, 1). Find
the coordinates of T.
Step 1 Let the coordinates of T equal (x, y).
Step 2 Use the Midpoint Formula:
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 2 Continued
Step 3 Find the x-coordinate.
Set the coordinates equal.
Multiply both sides by 2.
–2 = –6 + x
+ 6 +6
4=x
Simplify.
Add.
2 = –1 + y
+1 +1
Simplify.
3=y
The coordinates of T are (4, 3).
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Warm UP
Manuel is out for a jog. The thick
lines on the grid are jogging paths.
He is on his way home and is at D.
His home is at E. Each unit on the
grid is 1 mile.
1. Name the coordinates of D.
2. Find how many miles Manuel will jog if
he goes straight to the x-axis.
3. Find how many miles Manuel will jog if he stays on the
jogging paths all the way home.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
The Ruler Postulate can be used to find the distance
between two points on a number line. The Distance
Formula is used to calculate the distance between
two points in a coordinate plane.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 3: Using the Distance Formula
Find FG and JK.
Then determine whether FG  JK.
Step 1 Find the
coordinates of each point.
F(1, 2), G(5, 5), J(–4, 0),
K(–1, –3)
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 3 Continued
Step 2 Use the Distance Formula.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 3
Find EF and GH. Then determine if EF  GH.
Step 1 Find the coordinates of
each point.
E(–2, 1), F(–5, 5), G(–1, –2),
H(3, 1)
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 3 Continued
Step 2 Use the Distance Formula.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
You can also use the Pythagorean Theorem to
find the distance between two points in a
coordinate plane. You will learn more about the
Pythagorean Theorem in Chapter 5.
In a right triangle, the two sides that form the
right angle are the legs. The side across from the
right angle that stretches from one leg to the
other is the hypotenuse. In the diagram, a and b
are the lengths of the shorter sides, or legs, of the
right triangle. The longest side is called the
hypotenuse and has length c.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 4: Finding Distances in the Coordinate Plane
Use the Distance Formula and the
Pythagorean Theorem to find the distance, to
the nearest tenth, from D(3, 4) to E(–2, –5).
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 4 Continued
Method 1
Use the Distance Formula. Substitute the
values for the coordinates of D and E into the
Distance Formula.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 4 Continued
Method 2
Use the Pythagorean Theorem. Count the units for
sides a and b.
a = 5 and b = 9.
c2 = a2 + b2
= 52 + 92
= 25 + 81
= 106
c = 10.3
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 4a
Use the Distance Formula and the
Pythagorean Theorem to find the distance,
to the nearest tenth, from R to S.
R(3, 2) and S(–3, –1)
Method 1
Use the Distance Formula. Substitute the
values for the coordinates of R and S into the
Distance Formula.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 4a Continued
Use the Distance Formula and the
Pythagorean Theorem to find the distance,
to the nearest tenth, from R to S.
R(3, 2) and S(–3, –1)
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 4a Continued
Method 2
Use the Pythagorean Theorem. Count the units for
sides a and b.
a = 6 and b = 3.
c2 = a2 + b2
= 62 + 32
= 36 + 9
= 45
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 4b
Use the Distance Formula and the
Pythagorean Theorem to find the distance,
to the nearest tenth, from R to S.
R(–4, 5) and S(2, –1)
Method 1
Use the Distance Formula. Substitute the
values for the coordinates of R and S into the
Distance Formula.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 4b Continued
Use the Distance Formula and the
Pythagorean Theorem to find the distance,
to the nearest tenth, from R to S.
R(–4, 5) and S(2, –1)
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 4b Continued
Method 2
Use the Pythagorean Theorem. Count the units for
sides a and b.
a = 6 and b = 6.
c2 = a2 + b2
= 62 + 62
= 36 + 36
= 72
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 5: Sports Application
A player throws the ball
from first base to a point
located between third
base and home plate and
10 feet from third base.
What is the distance of
the throw, to the nearest
tenth?
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 5 Continued
Set up the field on a coordinate plane so that home
plate H is at the origin, first base F has coordinates
(90, 0), second base S has coordinates (90, 90), and
third base T has coordinates (0, 90).
The target point P of the throw has coordinates (0, 80).
The distance of the throw is FP.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 5
The center of the pitching
mound has coordinates
(42.8, 42.8). When a
pitcher throws the ball from
the center of the mound to
home plate, what is the
distance of the throw, to
the nearest tenth?
 60.5 ft
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Lesson Quiz: Part I
1. Find the coordinates of the midpoint of MN with
endpoints M(-2, 6) and N(8, 0). (3, 3)
2. K is the midpoint of HL. H has coordinates (1, –7),
and K has coordinates (9, 3). Find the coordinates
of L. (17, 13)
3. Find the distance, to the nearest tenth, between
S(6, 5) and T(–3, –4). 12.7
4. The coordinates of the vertices of ∆ABC are A(2, 5),
B(6, –1), and C(–4, –2). Find the perimeter of
∆ABC, to the nearest tenth. 26.5
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Lesson Quiz: Part II
5. Find the lengths of AB and CD and determine
whether they are congruent.
Holt Geometry