30. The sides of a square are 3 cm long. One vertex of the square is at (2,0) on a square coordinate grid marked in centimeter units. Which of the following

30. The sides of a square are 3 cm long. One vertex of the square is at (2,0) on a square coordinate grid marked in centimeter units. Which of the following points could also be a vertex of the square?F. (−4, 0)G. (0, 1)H. (1, −1)J. (4, 1)K. (5, 0)

(2, 0)(-4, 0)

distance = 6

Unsure where to start, start with the answers.

The distance is 6. I’m looking for a distance of 3. Wrong answer.

Plot and label everything (of course).

The Plan?

But now I know what the problem is asking for.


(0, 1)

(2, 0)

Wrong answer.


(2, 0)

distance = 3

(5, 0)

Right answer!

31. For ΔFGH, shown below, which of the following is an expression for y in terms of x ?

F

4

G

H

x

y

2 2 24y x

2 2 16y x

2 16y x

Pythagorean Theorem, of course.

The Plan?

32a. A bag contains 12 red marbles, 5 yellow marbles, and 15 green marbles. How many additional red marbles must be added to the 32 marbles already in the bag so that the probability of randomly drawing a red marble is 3/5? (choices are 13, 18, 28, 32, 40)

12

32p red

6

16

3

8

add 13: 12 13

32 13p red

25

45

5

9

add 18: 12 18

32 18p red

30

50

3

5

What % is Red now?

The Plan?

Nope

Yes!

Add some red marbles. How many? Try the answers!

32b. A bag contains 12 red marbles, 5 yellow marbles, and 15 green marbles. How many additional red marbles must be added to the 32 marbles already in the bag so that the probability of randomly drawing a red marble is 3/5? (choices are 13, 18, 28, 32, 40)

12 3

32 5

r

r

60 5 96 3r r

2 36r

18r

A Quicker Way (maybe)I like the first method many times because it helps me “get the feel

of the problem”.

33. What are the quadrants of the standard (x, y) coordinate plane below that contain points on the graph of the equation 4x − 2y = 8 ?

2 4y x

4 2 8x y

2 4 8y x

Slope-Intercept Form:

y mx b

slope y-intercept

Slope-Intercept Form - ALWAYS

The Plan?

quadrant

IV

quadrant

Iquadrant

II

quadrant

III

What’s the question? What quadrants does the line go in? Graph the line.

34. The graph of y = −5x2 + 9 passes through (1, 2a) in the standard (x, y) coordinate plane. What is the value of a ?

25 9y x

22 5 1 9a

2 5 9a

2 4a

2a

If a point is on the equation, it must

satisfy the equation.

If the line passes through the point, then the point must work in the equation.

The Plan?

34b. The graph of y = −5x2 + 9 passes through (1, 2a) in the standard (x, y) coordinate plane. What is the y-coordinate of the point?

25 9y x

22 5 1 9a

2 5 9a

2 4a

2a

If a point is on the equation, it must

satisfy the equation.

If the line passes through the point, then the point must work in the equation.

The Plan?

The same problem, though now asking for something different. Make sure you’re answering the question

asked!

(1, 4)

35. Jerome, Kevin, and Seth shared a submarine sandwich. Jerome ate 1 /2 of the sandwich, Kevin ate 1/3 of the sandwich, and Seth ate the rest. What is the ratio of Jerome’s share to Kevin’s share to Seth’s share?

Jerome1/2

Though it’s a submarine sandwich, it’s easier for me to think in terms of a pie.

Jerome at ½ of the “pie”.

I like drawing pies for fractions. Get a feel for the problem before charging in.

The Plan?

35. Jerome, Kevin, and Seth shared a submarine sandwich. Jerome ate 1 /2 of the sandwich, Kevin ate 1/3 of the sandwich, and Seth ate the rest. What is the ratio of Jerome’s share to Kevin’s share to Seth’s share?

Jerome: 3

Pieces of Pie

Kevin: 2Seth: 1

Ratio of Jerome to Kevin to Seth:

3:2:1

Kevin1/3

Jerome1/2

Seth?

Kevin ate 1/3 of the “pie”.

Seth ate the rest.

I like drawing pies for fractions. Get a feel for the problem before charging in.

The Plan?

Now make sure you’re answering the question asked.

36. A particular circle in the standard (x,y) coordinate plane has an equation of

(x − 5)2 + y2 = 36. What are the radius of the circle, in coordinate units, and the coordinates of the center of the circle?

Always try to graph to get started. Though this may look hard, make it look easy by setting something equal to 0.

The Plan?

What about the x-coordinate? How do I get y alone?

C(5,0)

r=6

what happens when y = 0

what happens when x = 5

25 0 36x

25 36x

5 6x

2 25 5 36y

2 36y

6y

11, 1x

36b. A thought on circles in general …

(x, y)

x

yrr2 = x2 + y2

The equation for a circle with Center (0,0) and Radius r :

1. Draw a circle

2. Find any point on that circle.

3. How far is that point from the origin?

Some Thoughts on the Circle

37. The figure below consists of a square and 2 semicircles, with dimensions as shown. What is the outside perimeter, in centimeters, of the figure?

8cm

8cm

The Plan?

Outline what you’re looking for, labeling everything.


8cm

8cm

The Plan?



8cm

8cm

8cm

The Plan?



8cm

8cm

8cm

The Plan?



8cm

8cm

8cm

The Plan?



8cm

8cm

8cm

4

2 48

2C r

4

8 4 8 4P

16 8

4

The Plan?

Outline what you’re looking for, labeling everything.You need the radius of the circle to find its circumference. How can you get it?

Remember to label everything!

38. In the figure below, points E and F are the midpoints of sides AD and BC of rectangle ABCD, point G is the intersection of AF and BE , and point H is the intersection of CE and DF. The interior of ABCD except for the interior of EGFH is shaded. What is the ratio of the area of EGFH to the area of the shaded region?

Triangles in EGFH

Shaded Triangles

2 6

Ratio = 2 : 6 = 1 : 3

The Plan?

1. This is a tricky one (for me). Because I don’t know how to solve it, what can I do with it?

2. Since it’s asking for the area of the shaded vs. the unshaded, part, and the shaded part is broken into triangles, what if I break the unshaded into triangles.

3. Assume all the triangles are equal area. Why? It’s better than doing nothing with the problem!

39. The coordinates of the endpoints of CD, in the standard (x, y) coordinate plane, are (−4, −2) and (14, 2). What is the x-coordinate of the midpoint of CD ?

The Plan?

1. Plot! Plot! Plot!


(14,2)

(-4, -2)

(5,0)

The Plan?

1. Plot! Plot! Plot!

2. Label everything! 3. Even if you know the formula, quickly find where the

midpoint is – about.


(6,1)(2,1) (4,1)

(4,1) is the midpoint of (2,1) and (6,1).

How did you get it?

Midpoint = 2

add numbers together

The Plan?

1. You don’t remember the formula, and approximating doesn’t help. What then?

2. Plot (and label) two points where you know the midpoint exactly, just by looking.

3. How do you get ‘4’ from ‘2’ and ‘6’? Add and divide by two.

40. What is the surface area, in square inches, of an 8-inch cube?

8

8A = 64

front & back 2 sides top & bottom

64 6totalArea

384

41. The equations below are linear equations of a system where a, b, and c are positive integers. ay + bx = cay − bx = c Which of the following describes the graph of at least 1 such system of equations in the standard (x, y) coordinate plane? I. 2 parallel linesII. 2 intersecting linesIII. A single line

ax by c

by ax c

a cy x

b b

ax by c

by ax c

a cy x

b b

a c

xb b

put equations in slope-intercept form

Since the slopes are different, the

lines intersect.

42. According to the measurements given in the figure below, which of the following expressions gives the distance, in miles, from the boat to the dock?

SOH CAH TOA

`

opp

adj

hyp

I need "opp" and have "adj".

tan52opp

adj

tan5230

opp

30 tan52 opp

53. The determinant of a matrix equals ad − cb. What must be the value of x for the matrix x

x 8x to have a determinant of −16 ?

28det 8

xx x

x x

8det 16

x

x x

2 8 16x x

2 8 16 0x x

4 4 0x x

4x

54. A formula for finding the value, A dollars, of P dollars invested at i% interest compounded annually for n years is A = P(1 + 0.01i)n. Which of the following is an expression for P in terms of i, n, and A ?

1 0.01n

A P i

1 0.01n

AP

i

58. What is the sum of the first 4 terms of the arithmetic sequence in which the 6th term is 8 and the 10th term is 13 ?

1 2 3 4 5 6 7 8 9 10138 10.59.256.755.54.2531.75

These are evenly spaced (like 0,5,10,15,20)

this is halfway between the 6th and 10th terms.

These are evenly spaced (like 0,5,10,15,20)

this is halfway between the 6th and 8th terms.

59. In the equation x2 + mx + n = 0, m and n are integers. The only possible value for x is –3. What is the value of m ?

x = -3 is the only solution.

x + 3 = 0

x + 3 is the only factor. 2 0x mx n

23 3x x x mx n

2 26 9x x x mx n

6 m

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30. The sides of a square are 3 cm long. One vertex of the square is at (2,0) on a square coordinate grid marked in centimeter units. Which of the following