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The Plan?

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)

Plot and label everything (of course).

Unsure where to start, start with the answers.

(-4, 0)

(2, 0)

distance = 6

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

But now I know what the problem is asking for.

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)

(0, 1)

(2, 0)

Wrong answer.

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)

(5, 0)

distance = 3

Right answer!

The Plan?

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

Pythagorean Theorem, of course.

H

y

4

G

x

F

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)

What % is Red now?

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

Nope

Yes!

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)

A Quicker Way (maybe)

I like the first method many times because it helps me “get the feel of the problem”.

The Plan?

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 ?

Slope-Intercept Form - ALWAYS

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

quadrant

II

quadrant

I

Slope-Intercept Form:

y-intercept

slope

quadrant

IV

quadrant

III

The Plan?

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

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

If a point is on the equation, it must satisfy 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?

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

If a point is on the equation, it must satisfy the equation.

(1, 4)

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

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?

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

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

Jerome at ½ of the “pie”.

Jerome

1/2

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?

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

Now make sure you’re answering the question asked.

Kevin ate 1/3 of the “pie”.

Seth ate the rest.

Seth

?

Pieces of Pie

Jerome:

3

Jerome

1/2

Kevin:

2

Kevin

1/3

Seth:

1

Ratio of

Jerome to Kevin to Seth:

3:2:1

The Plan?

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.

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

what happens when y = 0

what happens when x = 5

r=6

C(5,0)

Some Thoughts on the Circle

36b. A thought on circles in general …

1. Draw a circle

2. Find any point on that circle.

3. How far is that point from the origin?

(x, y)

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

r

y

r2 = x2 + y2

x

The Plan?

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?

Outline what you’re looking for, labeling everything.

8cm

8cm

The Plan?

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?

Outline what you’re looking for, labeling everything.

8cm

8cm

The Plan?

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?

Outline what you’re looking for, labeling everything.

8cm

8cm

8cm

The Plan?

Outline what you’re looking for, labeling everything.

8cm

8cm

8cm

The Plan?

Outline what you’re looking for, labeling everything.

8cm

8cm

8cm

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!

8cm

4

4

8cm

8cm

The Plan?

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?

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!

Triangles in EGFH

Shaded

Triangles

2

6

Ratio = 2 : 6

= 1 : 3

The Plan?

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 ?

1. Plot! Plot! Plot!

The Plan?

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 ?

1. Plot! Plot! Plot!

2. Label everything!

3. Even if you know the formula, quickly find where the midpoint is – about.

(14,2)

(5,0)

(-4, -2)

The Plan?

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 ?

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.

(4,1) is the midpoint of (2,1) and (6,1). How did you get it?

add numbers together

Midpoint =

2

(6,1)

(4,1)

(2,1)

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

8

A = 64

8

front & back

2 sides

top & bottom

41. The equations below are linear equations of a system where a, b, and c are positive integers.

ay +bx= c

ay −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 lines

II. 2 intersecting lines

III. A single line

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?

opp

hyp

adj

I need "opp" and have "adj".

SOH CAH TOA

`

53. The determinant of a matrix equals ad− cb. What must be the value of x for the matrix xx8x to have a determinant of −16 ?

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 isan expression for P in terms of i, n, and A ?

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 ?

These are evenly spaced

(like 0,5,10,15,20)

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

1.75

3

4.25

5.5

6.75

8

9.25

10.5

13

1

2

3

4

5

6

7

8

9

10

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

These are evenly spaced

(like 0,5,10,15,20)

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.