Sat - MC Grawhill part 39 doc

The Pythagorean TheoremThe Pythagorean theorem says that in any right triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side.. If you know t

6 E Label the two angles

that are “vertical” to those

marked b and c

No-tice that the

angle marked

a is an “external”

angle By the

External Angle theorem, a = b + c.

7 D Write in the missing angle measures by using

the fact that the sum of

angles in a triangle is

180° Then use the fact

that the biggest side of

a triangle is

always across from the

biggest angle to order

the sides of each

trian-gle

8 A Consider the side of length 4 to

be the base, and “attach” the side of length 6 Notice that the triangle has the greatest possible height when the two sides form a right angle

Therefore, the greatest possible area

of such a triangle is (1/2)(4)(6) = 12, and the minimum possible area is 0

The greatest prime number less than 12 is 11

30°

35°

60°

65°

C D

55 °

115°

6

4

a°

b°

c°

The Pythagorean Theorem

The Pythagorean theorem says that in any right

triangle, the sum of the squares of the two

shorter sides is equal to the square of the longest

side If you know two sides of any right

trian-gle, the Pythagorean theorem can always be

used to find the third side

Example:

In the figure below, what is x?

Pythagorean theorem: 92+ x2= 152

You can also use the modified Pythagorean

theorem to find whether a triangle is acute or

obtuse

If (side1)2+ (side2)2< (longest side)2, the triangle is

obtuse (If the stick gets bigger, the alligator’s

mouth gets wider!)

If (side1)2+ (side2)2> (longest side)2, the triangle is

acute (If the stick gets smaller, the alligator’s

mouth gets smaller!)

Special Right Triangles

Certain special right triangles show up frequently on

the SAT If you see that a triangle fits one of these

pat-terns, it may save you the trouble of using the

Pythagorean Theorem But be careful: you must

know two of three “parts” of the triangle in order to

assume the third part

x

15 9

c2

c

b2

a2

a b

a2+ b2= c2

3-4-5 triangles More accurately, these can

be called 3x-4x-5x triangles because the

multi-ples of 3-4-5 also make right triangles Notice that the sides satisfy the Pythagorean theorem

5-12-13 triangles Likewise, these can be

called 5x-12x-13x triangles because the multi-ples of 5-12-13 also make right triangles

No-tice that the sides satisfy the Pythagorean theorem

45 °-45°-90° triangles These triangles can be

thought of as squares cut on the diagonal This

shows why the angles and sides are related the way they are Notice that the sides satisfy the Pythagorean theorem

x x 2

45°

45°

x

5

12 13 2.5

6 6.5

3 4 5

16

12 20

Lesson 3: The Pythagorean Theorem

The Distance Formula:

d2= (x2− x1)2+ ( y2− y1)2 so

y

x

(x2, y2)

(x1, y1)

O

x2– x1

y2– y1

d

d= (x2−x1) +(y −y)

2

2

30 °-60°-90° triangles These triangles can

be thought of as equilateral triangles cut in half.

This shows why the angles and sides are

re-lated the way they are Notice that the sides

satisfy the Pythagorean theorem

The Distance Formula

Say you want to find the distance between two points

(x1, y1) and (x2, y2) Look carefully at this diagram

and notice that you can find it with the Pythagorean

theorem Just think of the distance between the

points as the hypotenuse of a right triangle, and the

Pythagorean theorem becomes—lo and behold—

the distance formula!

60°

30°

2x

x

x 3

Concept Review 3: The Pythagorean Theorem

1 Draw an example of each of the four “special” right triangles

Use the modified Pythagorean theorem and the triangle inequality to find whether a triangle with the given side lengths is acute, obtuse, right, or impossible

8 The circle above has its center at P and an area of 16 π If AP = AB, what is the area of ΔABC?

9 The area of the triangle above is 30 What is the value of h?

10 What is the height of an equilateral triangle with sides of length cm?

11 Point P is at (0, 0), point M is at (4, 0), and point N is at (9, 12) What is the perimeter of ΔMNP?

6 3

5

h

C

A B P

1. The length and width of a rectangle are in the

ratio of 5:12 If the rectangle has an area of 240

square centimeters, what is the length, in

cen-timeters, of its diagonal?

(A) 26

(B) 28

(C) 30

(D) 32

(E) 34

2. A spider on a flat horizontal surface walks

10 inches east, then 6 inches south, then

4 inches west, then 2 inches south At this

point, how many inches is the spider from its

starting point?

(A) 8

(B) 10

(C) 12

(D) 16

(E) 18

3. In the figure above, ABCF is a square and

ΔEFD and ΔFCD are equilateral What is the

measure of ∠AEF?

(A) 15°

(B) 20°

(C) 25°

(D) 30°

(E) 35°

C

D E

F

4. In the figure above, an equilateral triangle is drawn with an altitude that is also the diameter

of the circle If the perimeter of the triangle is 36, what is the circumference of the circle?

(A) (B) (C) (D) (E) 36π

5. In the figure above, A and D are the centers of the two circles, which intersect at points C and

E CE –– is a diameter of circle D If AB = CE = 10, what is AD?

(A) 5 (B) (C) (D) (E) 10 3

10 2

5 3

5 2

C A

B E D

12 3π

12 2π

6 3π

6 2π

SAT Practice 3: The Pythagorean Theorem

6. Point H has coordinates (2, 1), and point J

has coordinates (11, 13) If HK _is parallel to

the x-axis and JK _ is parallel to the y-axis, what

is the perimeter of ΔHJK?

7. A square garden with a diagonal of length

meters is surrounded by a walkway

3 meters wide What is the area, in square

meters, of the walkway?

1

2

3

4

5

7

8

6

1

0

2

3

4

5

7

8

6

1 0

2 3 4 5

7 8 6

1 0

2 3 4 5

7 8 6

24 2

24 2 3

Note: Figure not drawn to scale

8. In the figure above, what is the value of z?

(A) 15 (B) (C) (D) (E) 30 3

30 2

15 3

15 2

x°

x°

2x°

2y + 5

1

2

3

4

5

7

8

9

6

1

0

2

3

4

5

7

8

9

6

1 0

2 3 4 5

7 8 9 6

1 0

2 3 4 5

7 8 9 6

Concept Review 3

1 Your diagram should include one each of a

3x-4x-5x, a 5x-12x-13x, a 30 °-60°-90°, and a

45°-45°-90° triangle.

2 Obtuse: 52+ 62= 61 < 92= 81

3 Acute: 22+ 122= 148 > 122= 144

4 Obtuse: 62+ 82= 100 < 112= 121

5 Impossible: 2 + 2 isn’t greater than 12

6 Impossible: 3 + 4 isn’t greater than 7

7 Right: 1.52+ 22= 6.25 = 2.52

8 Since the area of a circle is

πr2= 16π, r = 4 Put the

infor-mation into the diagram

Use the Pythagorean

theorem or notice that,

since the hypotenuse is twice

the shorter side, it is a 30°-60°-90° triangle

Either way, , so the area of the triangle

is ( )bh 2=( )4 4 3( ) 2 8 3=

CB= 4 3

9 At first, consider the shorter leg as the base In this case, the other leg is the

height Since the area is

(bh)/2 = 30, the other leg must be 12 This is a 5-12-13 triangle,

so the hypotenuse is 13 Now consider the

hypotenuse as the base Since 13h/2 = 30,

h= 60/13 = 4.615

10 Your diagram should look like this: The height is

11 Sketch the diagram Use the Pythagorean

theo-rem or distance for-mula to find the lengths The perimeter is

4 + 13 + 15 = 32

3 3 3 9

( )( )=

Answer Key 3: The Pythagorean Theorem

C

A

B P

4 4 4

4 3

5

h

12

13

3 3 9

4

N(9, 12)

12

5

13 15

y

x

SAT Practice 3

1 A Draw the rectangle If the length and width are

in the ratio of 5:12, then they can be expressed as

5x and 12x The area, then, is (5x)(12x) = 60x2= 240

So x= 2, and the length and width are 10 and 24

Find the diagonal with the Pythagorean theorem:

102+ 242= a2, so 100 + 576 = 676 = a2and d= 26

(Notice that this is a 5-12-13 triangle times 2!)

2 B Draw a diagram like this

The distance from the starting

point to the finishing point is

the hypotenuse of a right triangle

with legs of 6 and 8 Therefore, the

distance is found with Pythagoras:

62+ 82= 36 + 64 = 100 = d2, so d= 10

(Notice that this is a 3-4-5 triangle times 2!)

3 A Draw in the angle measures

All angles in a square are 90° and

all angles in an equilateral

tri-angle are 60° Since all of the

angles around point F add up

to 360°, ∠EFA = 360 – 60 − 60 −

90 = 150° Since EF = AF, ΔEFA

is isosceles, so ∠AEF = (180 −

150)/2 = 15°

4 B If the perimeter of the triangle is 36, each side must have a length

of 12 Since the altitude forms two 30°-60°-90°

triangles, the altitude must have length This is also the diameter

of the circle The circum-ference of a circle is π times the diameter:

5 C Draw in AE and AC Since

all radii of a circle are equal, their measures are both 10

as well Therefore ΔACE is equilateral, and AD divides it

into two 30°-60°-90° triangles

You can use the Pythagorean theorem, or just use the 30°-60°-90° relationships

to see that AD= 5 3

6 3π

6 3

10

6 4 2

6 8 Start

Finish

C

D E

F

60° 60°

60 °

60°

60°

60°

90 °

90 °

90 °

150 °

6 6

60°

60°

6 3

C A

B E D

10 10 10 5 5

6 36 Sketch a diagram Point K has coordinates

(11, 1) ΔHJK is a right triangle, so it satisfies the

Pythagorean theorem Your diagram should look

like this one The perimeter is 9 + 12 + 15 = 36

7 324 Since the garden is a

square, the diagonal divides it into 45°-45°-90° triangles

Therefore the sides have a length of 24 The outer edge of the walkway is therefore

24 + 3 + 3 = 30 The area of the walkway is the difference of the areas of the squares:

302− 242= 324

8 B The sum of the angles is 180°, so x + x + 2x = 4x = 180, and x = 45 Therefore the triangle is a

45°-45°-90° triangle Since it is isosceles, 3y = 2y + 5, and therefore y= 5 The three sides, then, have lengths of 15, 15, and 15 2

24 2 3

30 24

H(2, 1)

J(11, 13)

K(11, 1) x

y

12 9

15

Working with Slopes

Every line has a slope, which is the distance you

move up or down as you move one unit to the

right along the line Imagine walking between

any two points on the line As you move, you go

a certain distance “up or down.” This distance

is called the rise You also go a certain distance

“left or right.” This distance is called the run

The slope is simply the rise divided by the run

Slope = rise

run

= 2−− 1

Plotting Points

Some SAT questions may ask you to work with points

on the x-y plane (also known as the coordinate plane

or the Cartesian plane, after the mathematician and

philosopher René Descartes) When plotting points,

remember these four basic facts to avoid the common

mistakes:

• The coordinates of a point are always

given alphabetically: the x-coordinate

first, then the y-coordinate.

• The x-axis is always horizontal and the

y-axis is always vertical.

• On the x-axis, the positive direction is to the

right of the origin (where the x and y axes

meet, point (0,0))

• On the y-axis, the positive direction is up

from the origin (where the x and y axes

meet, point (0,0))

You should be able to tell at a glance whether

a slope has a positive, negative, or zero slope If

the line goes up as it goes to the right, it has a positive slope If it goes down as it goes to the right, it has a negative slope If it’s horizontal,

it has a zero slope

If two lines are parallel, they must have the same slope

y

x

(3, 5)

3 "right"

O

5 "up"

y

x

(x2, y2)

(x1, y1)

O

run = x2− x1

rise = y2− y1

y

x O

positive slope

negative slope

0 slope

Finding Midpoints

The midpoint of a line segment is the point that divides the segment into two equal parts

Think of the midpoint as the average of the two endpoints

Midpoint = x1 x2 y1 y2

⎛

⎝⎜

⎞

⎠⎟

,

y

x

(x2, y2)

(x1, y1)

O

(x1+x2,y1+y2

)

2 2

Lesson 4: Coordinate Geometry

Concept Review 4: Coordinate Geometry

Questions 1–10 refer to the figure below

Horizontal and vertical lines are spaced 1 unit apart

1 What are the coordinates

2 What are the coordinates of the

3 What is the slope of AB ––?

4 Draw a line through B that is parallel to the

y-axis and label it 艎1

5 What do all points on 艎1have

x

y

A

B

6 Draw a line through B that is parallel to the x-axis

and label it l2

7 Draw the line y= −1 and label it 艎3

8 If point A is reflected over 艎2, what are the

9 If line segment AB ––is rotated 90° clockwise about

point B,what are the coordinates of the image of

point A?

10 If point B is the midpoint of line segment AC ––,

what are the coordinates of point C?

Rectangle ABCD has an area of 108.

Note: Figure not drawn to scale

Questions 11–15 pertain to the figure above

11 k = _ m = _ n = _ p = _

12 What is the ratio of AC to the perimeter of ABCD?

13 What is the slope of DB –––?

14 At what point do AC ––and

15 If B is the midpoint of segment DF –––, what are the

coordinates of point F?

x

y A(2, k) B(m, n)

C(14, p) D(2, 1)

O

Questions 16–18 pertain to the figure above

16 What is the area of the

17 If the triangle above were reflected over the line

x = 3, what would be the least x-coordinate of any

point on the triangle?

18 If the triangle above were reflected over the line

y= 1, what would the area of the

x

y

(6, 7)

(-1, 3)

(-1, -1)