1006 formulas and geom

Holt McDougal Geometry

1-5 Using Formulas in Geometry

Drill #5 9/11/14Evaluate. Round to the nearest hundredth.

1. 122

2. 7.62

3.

4.

5. 32()

6. (3)2

144

57.76

8

7.35

28.27

88.83



Apply formulas for perimeter, area, and circumference.

Objective



Develop and apply the formula for midpoint.

Use the Distance Formula and the Pythagorean Theorem to find the distance between two points.

Objectives



perimeter diameterarea radiusbase circumferenceheight pi

Vocabulary



• Make a list of any math formula you know and be able to explain how to use it.



1. Graph A (–2, 3) and B (1, 0).

2. Find CD. 8

3. Find the coordinate of the midpoint of CD. –2

4. Simplify.

4



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.



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.



Perimeter is expressed in linear units, such as inches (in.) or meters (m). Area is expressed in square units, such as square centimeters (cm2).

Remember!



Find the perimeter and area of each figure.

Example 1A: Finding Perimeter and Area

= 2(6) + 2(4)

= 12 + 8 = 20 in.

= (6)(4) = 24 in2



Find the perimeter and area of each figure.

Example 1B: Finding Perimeter and Area

= (x + 4) + 6 + 5x

P = a + b + c

= 6x + 10= 3x + 12



Check It Out! Example 1

P = 4s

Find the perimeter and area of a square with s = 3.5 in.

A = s2

P = 4(3.5) A = (3.5)2

P = 14 in. A = 12.25 in2



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.

Example 2: Crafts Application

The area of one triangle is

The total area of the 12 triangles is 12(8) = 96 in2.




Find the amount of fabric used to make four

rectangles. Each rectangle has a length of in.

and a width of in.

The area of one triangle is

The amount of fabric to make four rectangles is



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.



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 .



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 201.1 cm2




Find the circumference and area of a circle with radius 14m.

88.0 m 615.8 m2



Lesson Quiz: Part I

Find the area and perimeter of each figure.

1. 2.

3.

23.04 cm2; 19.2 cm

x2 + 4x; 4x + 8

10x; 4x + 16



Lesson Quiz: Part II

Find the circumference and area of each circle. Leave answers in terms of .

4. radius 2 cm

5. diameter 12 ft

6. The area of a rectangle is 74.82 in2, and the

length is 12.9 in. Find the width.

4² cm; 4 cm2

36 ft 2; 12 ft

5.8 in



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).



To make it easier to picture the problem, plot the segment’s endpoints on a coordinate plane.

Helpful Hint



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)




Find the coordinates of the midpoint of EF with endpoints E(–2, 3) and F(5, –3).



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:



Example 2 Continued

Step 3 Find the x-coordinate.

Set the coordinates equal.

Multiply both sides by 2.

12 = 2 + x Simplify.

– 2 –2

10 = x

Subtract.

Simplify.

2 = 7 + y– 7 –7

–5 = y

The coordinates of Y are (10, –5).




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:



Check It Out! Example 2 Continued

Step 3 Find the x-coordinate.

Set the coordinates equal.

Multiply both sides by 2.

–2 = –6 + x Simplify.

+ 6 +6

4 = x

Add.

Simplify.

2 = –1 + y+ 1 + 1

3 = y

The coordinates of T are (4, 3).



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.



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)



Example 3 Continued

Step 2 Use the Distance Formula.




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)



Check It Out! Example 3 Continued

Step 2 Use the Distance Formula.



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.



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).



Example 4 Continued

Method 1Use the Distance Formula. Substitute thevalues for the coordinates of D and E into theDistance Formula.



Method 2Use the Pythagorean Theorem. Count the units for sides a and b.

Example 4 Continued

a = 5 and b = 9.

c2 = a2 + b2

= 52 + 92

= 25 + 81

= 106

c = 10.3



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 1Use the Distance Formula. Substitute thevalues for the coordinates of R and S into theDistance Formula.



Check It Out! Example 4a Continued


R(3, 2) and S(–3, –1)




a = 6 and b = 3.

c2 = a2 + b2

= 62 + 32

= 36 + 9

= 45

Check It Out! Example 4a Continued



Check It Out! Example 4b


R(–4, 5) and S(2, –1)

Method 1Use the Distance Formula. Substitute thevalues for the coordinates of R and S into theDistance Formula.



Check It Out! Example 4b Continued


R(–4, 5) and S(2, –1)




a = 6 and b = 6.

c2 = a2 + b2

= 62 + 62

= 36 + 36

= 72

Check It Out! Example 4b Continued



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?

Example 5: Sports Application



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.

Example 5 Continued



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



Lesson Quiz: Part I

(17, 13)

(3, 3)

12.73. Find the distance, to the nearest tenth, between

S(6, 5) and T(–3, –4).

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

1. Find the coordinates of the midpoint of MN with endpoints M(-2, 6) and N(8, 0).

2. K is the midpoint of HL. H has coordinates (1, –7), and K has coordinates (9, 3). Find the coordinates of L.



Lesson Quiz: Part II

5. Find the lengths of AB and CD and determine whether they are congruent.

1006 formulas and geom

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