Math B Assignments: Introduction to Proofs



Geometry Assignments: Coordinate Geometry

|Day |Topics |Homework |HW Grade |Quiz Grade |

|1 |Slope formula |HW CG - 1 | | |

|2 |Equations of lines, graphing lines |HW CG - 2 | | |

|3 |Parallel and perpendicular lines |HW CG - 3 | | |

|4 |Distance formula ***QUIZ*** |HW CG - 4 | | |

|5 |Equations of circles |HW CG - 5 | | |

|6 |Completing the square |HW CG - 6 | | |

|7 |Midpoint formula |HW CG - 7 | | |

|8 |Dividing a segment in a given ratio |HW CG - 8 | | |

|9 |Using the formulas |HW CG - 9 | | |

|10 |Coordinate geometry proofs ***QUIZ*** |HW CG - 10 | | |

|11 |Review |HW CG - Review | | |

| |**TEST** | | | |

HW CG – 1

1. 3/2, –1/3, 0 and undefined

2a. –2/3 b. 0 c. undef. d. –2

3. x = 3 4. k = –2 5. a = –2 ( a = 5

6a. Points all lie on the same line b. They must all be the same. 2) Same

6. No; slope of [pic] = –7/12 ( slope of [pic]=–4/7 7. x = 3.5

8. 4/3 9a. 17.5' b. 18.2' 10. 6.67'

HW CG – 2

1a. m = 0, b = 5 b. m = –2, b = 0 c. m = –1, b = 8 d. m = 1/2, b = –2

3a. y = –2x + 6 b. y = [pic] c. y = 4

4. y = 3x –15 5. y = –2x + 8 6. x = 3

7b. y increases by 3. d. y decreases by 1/2 e. C’mon, figure it out.

HW CG – 3

1a. 3/4 b. 3/4 c. –4/3 2a. No b. Yes 3. (A no; (D yes

4. [pic] 5. [pic] 6. [pic]

HW CG – 4

1a. 60 b. 80 c. 100 d. [pic]

2a. a. 10π b. 31.42 c. 25π d. 78.54

3b. [pic] b. 8 4c. 43.42 6d. They must measure perpendicular to the line f. [pic]

HW CG – 5

1. O: center (0, 0), r = 4, x2 + y2 = 16 C: center (6, –1), r = 3, (x – 6)2 + (y + 1)2 = 9

2a. 6, (0, 0) b. , (3, –12) c. 12 , (2, 0)

3a. x2 + y2 = 64 b. (x + 2)2 + (y – 5)2 = 30 4a. x2 + y2 = 169 b. yes, no, yes

5a. (x + 1)2 + (y – 3)2 = 20 b. 62.8 c. 28.1 6. (3, –5) ; [pic]

HW CG – 6

1. (–5, 4); 6 2. (0, 5/2); 9/2 3. What did you get for r? 4. Vertex (6, –12) is a min.

5. Vertex (–4, 7) is a max. 7a. 7 b. 3 c. 289 d. [pic]

HW CG – 7

1a. (5, 9) b. (23, 14.5) c. (3a, 5b) 2. (10, 2) 3. (–5, –6) 4. (1, 5.5)

5a1. [pic] a2. [pic] c. BOTH 6. y = –3x + 15

HW CG – 8

1. (1, 4) 2. (1, 3) and (5, 5) 3. y – 4 = –2(x – 3) 4a. (x + 2)2 + (y – 4)2 = 65 c. (5, 0) and (5, 8)

HW CG – 9

1a. Yes b. No c. No

3a. [pic] b. (a + 3b, a) c. [pic] and [pic]; slopes are opp. recip.

4a. 10 b. 50

HW CG – 10

1a. AC = BC = [pic] b. D(6, 3) c. [pic]and [pic]; opp. recip. slopes

2a. M(8, 3) and N(5, 4) b. [pic]; same slope c. [pic] and [pic]; [pic]

3a. [pic] and [pic]; opp. recip. slopes

b. M(3, 2) is midpoint[pic]; [pic] and [pic]; [pic]

Review Answers

1. –c/2 2. (–2a, 4b) 3. (–1, 5) 4. 8 5. [pic]

6. (2) 7. [pic] 8. None 9. (–6, 4); 3

10a. [pic], [pic] so [pic] b/c slopes are opp. recip. which makes (C a rt. ( and (ABC a rt. (.

b. Midpoint of [pic] is M(1, 2). AB = 10; CM = 5so [pic]

11a. AC = BC = b. (3, 2) c. [pic] d. Slope of [pic]= 1/3; slope of [pic]= –3/1

12. a. Opposite slopes either 0 or s/t. b. AB = s; AD = [pic]

13a. 100.56 kPa b. Decreases by 10.36 kPa 10. (0, –7) 11. x = –3 12. (2, –1)

Geometry HW: CG - 1

Name

1. Find the slope of each side of quadrilateral ABCD shown

in the figure at right.

mAB = mBC =

mCD = mAD =

2. Find the slope of the line segment joining each pair of points.

a. (–23, 39) and (58, –15) b. (14, 32) and (–27, 32)

c. (–36, 24) and (–36, 55) d. (a, 3a – b) and (a + 2b, 3a – 5b)

3. Find the value of x so that the line passing through the points (3, –2) and (x, 6) will have an undefined slope.

4. Find the value of k so that the line passing through (3, –2) and (3k + 5, k – 6) will have a slope of 3/2.

5. Find two values of a so that the line passing through (a, 10) and (7, a2 – 3a) will have a slope of 0.

6. Determine if the three points R(–7, –5), S(5, 2) and T(12, 6) are collinear. Justify your answer. (Think: If all three are on the same line, what must be true about the slopes of [pic] and [pic]?)

7. Find the value of x that will make the points J(–4, 15), K(x, 10) and L(14, 3) collinear.

8. A ladder 15 feet long leans against a vertical wall. The top of the ladder is 12 feet above the level ground. What is the slope of the ladder (assume it’s positive)?

9. Tommy Hawk is building a skateboard ramp. He wants it to have a slope of 2/7 and a vertical rise (height) of 5 feet.

a. What horizontal distance will the ramp cover?

b. How long will the actual ramp be?

10. A certain roof has a pitch (a builder’s word for slope) of 5/12 on each side. The entire roof is to be 32 feet wide. How high will the ridge line be above the attic floor?

Geometry HW: CG - 2

Name

1. Find the slope and y-intercept for each of the following lines. Then graph each line.

a. y = 5

b. y = –2x

c. y = 8 – x

d. 3x – 6y = 12

2. On a new set of axes, graph and label the following:

a. y ( x + 1

b. 2x + 3y < 12

c. x ( 6

3. Write the equation of the line having the given slope and y-intercept:

a. slope = –2, y-intercept is 6 b. slope = ; y-intercept at the origin c. slope = 0, y-intercept is 4

4. Find the equation of the line having slope 3 and passing through the point (4, –3).

5. Find the equation of the line that passes through the points (3, 2) and (6, –4).

6. Find the equation of the line passing through the points (3, –2) and (3, 4).

7. a. Graph the line y = 3x – 7.

b. For the line in part (a), how much does y change when x increases by 1 unit? Does y increase or decrease?

c. Graph the line [pic]. (This may go on the same axes as part a.)

d. For the line in part (c), how much does y change when x increases by 1 unit?

e. For the line [pic], how much does y change when x increases by one unit? Does y increase or decrease? (Note: you should be able to answer this without needing to graph the line.)

8. The speed of sound at sea-level depends on temperature according to the equation

S = 0.60T + 331.45 where S is the speed in meters per second and T is the temperature in degrees Celsius.

a. What is the slope of the line?

b. What is the speed of sound at 0°C?

c. Every time the temperature goes up by 1°C, by how much will the speed of sound change? Will it

increase or decrease?

Geometry HW: CG - 3

Name

1. a. Find the slope of the line 3x – 4y = 8.

b. Find the slope of a line parallel to the line in part a.

c. Find the slope of a line perpendicular to the line in part a.

2. Determine using slopes whether or not the two segments shown are parallel and give a specific reason why or why not.

a. b.

3. In the quadrilateral at right, determine using slopes if (A and/or (D are right angles. For each angle, give a specific reason why or why not.

4. Find the equation of a line parallel to the line 3x + 2y = 12 and passing through the point (6, –2).

5. Find the equation of a line perpendicular to the line [pic] and passing through the point (5, –4).

6. Two perpendicular lines have the same y-intercept. The equation of one of the lines is 2x + 3y = 12. Find an equation for the other line.

7. Tom has a line of slope 2/3. Sawyer has a line parallel to Tom’s with a slope of p/q. Must p = 2? Explain.

8. Triangle ABC has vertices A(–2, 3), B(6, 3) and C(6, 9).

a. Graph (ABC.

b. Find the area of the triangle.

c. Find the perimeter of the triangle.

Geometry HW: CG - 4

Name

Show appropriate work.

1. Find the distance between each pair of points:

a. (25, 72) and (85, 72) b. (–16, –30) and (–16, 50)

c. (45, 24) and (–35, 84) d. (a, b) and (a + b, 3b)

2. The coordinates of the endpoints of a diameter of a circle are P(–1, 4) and Q(7, –2).

a. Find the circumference of the circle in terms of π. (C = πd = 2(r. )

b. Find the circumference of the circle to the nearest hundredth.

c. Find the area of the circle in terms of π. (A = πr2. )

d. Find the area of the circle to the nearest hundredth.

3. a. Graph the lines y = 2, x = 6 and y = x on one set of axes.

b. Find the perimeter of the triangle formed by the three lines in part (a).

c. Find the area of the triangle formed by the three lines in part (a).

4. a. Graph the lines 3x – 4y = –4, x + 2y = –8, and

x = 8. The three lines should form a triangle. Label the leftmost vertex A, the uppermost vertex B and the remaining vertex C.

b. Show that ΔABC is an isosceles triangle. Show your work and give a written conclusion and reason of the form "ΔABC is isosceles

because . . . " Be specific.

c. Find the perimeter of ΔABC to the nearest hundredth.

5. a. Find the distance from the point (2, 5) to the line y = 1.

b. Find the distance from the point (2, 5) to the line x = 6.

6. We want to find the distance from the point (2, 5) to the line ( having equation[pic].

a. Graph the line and the point.

b. Val measured the distance from the point to the line vertically. What did she get?

c. Hal measured the distance from the point to the line horizontally. What did he get?

d. Explain why neither Hal nor Val got the correct distance from the point to the line. How should the distance from a point to a line be measured? (This is important. Remember it.)

e. Draw the line through (2, 5) that is perpendicular to line (.

f. What is the distance from the point (2, 5) to line (?

Geometry HW: CG – 5

Name

1. a. For circle O in the diagram at right, write the

coordinates of the center

length of the radius

equation of the circle

b. . For circle C in the diagram at right, write the

center radius equation

2. For each of the following circles, find the length of the radius and the coordinates of the center:

a. x2 + y2 = 36 Radius = Center:

b. (x – 3)2 + (y + 12)2 = 20 Radius = Center:

c. (x – 2)2 + y2 = 122 Radius = Center:

3. Write equations for the following circles:

a. Center at the origin; radius 8

b. Center at (–2, 5); radius

4. a. Write an equation of the set of all points that are 13 units from the origin.

b. Tell if each of the following points is in the set from part a:

1) (0, 13) 2) (6, 7) 3) (–5, 12)

5. a. Write the equation of the circle having a diameter with endpoints (–5, 1) and (3, 5).

b. Find the area of the circle to the nearest tenth.

c. Find the circumference of the circle to the nearest tenth.

Note: A compass would be helpful for these last two problems. If you are in school, see your favorite math teacher.

6. Solve the following system of equations graphically:

(x – 3)2 + y2 = 25

[pic]

7. a. Graph (RAT having vertices R(–4, 0), A(0, 8) and T(11, –3).

b. The point C(4, 1) is called the circumcenter of the circle (more on that later in the course). Show that C is equidistant from all three vertices of (RAT. Call that distance r.

c. Write the equation of the circle having its center at C and radius r. Graph the circle. What is special about this circle?

Geometry HW: CG – 6

Name

1. Find the coordinates of the center and the length of the radius of the circle x2 + 10x + y2 – 8y + 5 = 0.

2. Find the coordinates of the center and the length of the radius of the circle x2 + y2 – 5y –14 = 0.

3. Explain why the equation x2 – 2x + y2 + 6y + 14 = 0 does not represent the equation of a circle.

4. Use completing the square to write the quadratic function [pic] in vertex form [pic]. Give the coordinates of the vertex. Is it a maximum or a minimum for the function? How do you know?

5. Use completing the square to write the quadratic function [pic] in vertex form [pic].* Give the coordinates of the vertex. Is it a maximum or a minimum for the function? *There are a couple of ways to do this. The easiest may be to first divide everything by –2 so that x2 has a coefficient of 1. Then at the end, multiply everything by –2 to get y by itself again.

6. a. Graph (BUG having vertices B(–4, 1), U(8, 1) and T(8, 10).

b. The point I(5, 4) is called the incenter of the circle (more on that later in the course). Show that I is equidistant from sides [pic] and [pic]. Call that distance r. (I is also the same distance from the third side, [pic], but that is harder to figure out.)

c. Write the equation of the circle having I as its center and radius r. Graph the circle (a compass would be helpful). What is special about this circle?

7. a. What number is halfway between 4 and 10 on a number line?

b. What number is halfway between –2 and 8 on a number line?

c. What number is halfway between 125 and 453 on a number line?

d. What number is halfway between x1 and x2 on a number line?

Geometry HW: CG - 7

Name

Note: There is scratch graph paper on the back of this homework. There is more graph paper at the mailboxes

if that is not enough.

1. Find the coordinates of the midpoint of the segment that joins each pair of points:

a. (6, 8) and (4, 10) b. (58, –65) and (–12, 94) c. (5a, 2b) and (a, 8b)

2. M(7, 4) is the midpoint of [pic]. If the coordinates of C are (4, 6), find the coordinates of D.

3. The midpoint of [pic]is M(–1, 6). The coordinates of P are (x, y) and the coordinates of Q are

(x + 8, –3y). Find the values of x and y.

4. Segment [pic] has A(–2, 8) and B(10, –2). Find the coordinates of point Q on [pic]such that [pic].

5a. Give an appropriate conclusion for each of the following.

1) [pic] bisects [pic] at M

2) [pic] bisects [pic] at M

b. Which of the conclusions from part (a) would be true if [pic] and [pic] bisect each other at M ?

6. Segment [pic] has endpoints A(1, 2) and B(7, 4). Find the equation of the perpendicular bisector of [pic].

7. Verify using coordinate geometry that the line l with equation[pic] is the perpendicular bisector of the segment [pic] with endpoints A(–1, 7) and B(5, 3). (Note: this problem has two separate parts: perpendicular and bisector. Proving one does not automatically prove the other.)

Geometry HW: CG - 8

Name

1. Find the coordinates of the point P on the directed line segment from A(–8, 10) to B(13, –4) that partitions the segment into a ratio of 3:4.

2. Find the coordinates of the points P and Q that divide the segment from J(–3, 1) to K(9, 7) into three congruent parts. (What two ratios are implied here?)

3. Write the equation of the line that is the perpendicular bisector of [pic] with J(–3, 1) and K(9, 7).

4. a. Write the equation of the circle having center (–2, 4) and radius [pic].

b. Does the point (–8, 9) lie on the circle? Justify your answer.

c. Find two points on the line x = 5 that lie on the circle.

5. a. Graph (ABC having vertices A(0, 4), B(4, 14) and C(8, 0).

b. Find the midpoints of [pic], [pic] and [pic].

Call them M, N and P respectively.

c. Draw [pic], [pic] and [pic].

These are called medians of the triangle.

d. Find the point where all three medians intersect. Call it G.

This is called the centroid of the triangle.

e. Show that G divides medians [pic] and [pic] in a 2:1 ratio.

(It also divides median [pic] in the same ratio but you don’t

have to show that.)

Geometry HW: CG - 9

Name

Show work.

1. Quadrilateral ABCD has vertices A(–1, 1), B(2, 3), C(6, 0) and

D(3, –2). Determine using coordinate geometry whether or not

the diagonals* of ABCD

a. bisect each other.

b. are congruent.

c. are perpendicular.

Show work and give a reason for each of your answers.

*Diagonals in a quadrilateral connect opposite vertices (angles). The diagonals of ABCD are [pic] and [pic]. You need to know this.

2. Triangle ABC has vertices A(–1, –2), B(3, 6), and

C(11, 2). Show using coordinate geometry that (ABC is an isosceles right triangle. (Note: there are two parts to this problem, isosceles and right.)

3. Given the points X(0, 3b), Y(a, 0), and Z(a + 6b, 2a);

a. Find the length of [pic].

b. Find the midpoint of [pic].

c. Show that [pic]⊥ [pic].

You must know these definitions:

A median of a triangle is a line segment that goes from one vertex of the triangle to the midpoint of the opposite side. In the figure, M is the midpoint of [pic], so [pic] is a median of (ABC.

An altitude of a triangle is a line segment that starts from one vertex and is perpendicular to the opposite side (or to the line that contains that side). In the figure, [pic] so [pic] is an altitude of (ABC.

4. The vertices of ΔRST are R(11, –1), S(13, 10), and

T(3, 5).

a. Find the length of the median from S to [pic].

b. Show that the median from S to [pic]is also an altitude of the triangle.

c. Find the area of ΔRST.

Geometry HW: CG - 10

Name

1. Triangle ABC has vertices A(4, 0), B(8, 6), and C(0, 7).

a. Prove that (ABC is isosceles.

b. Find the coordinates of point D on [pic] so that [pic] is a median. (This part is not a proof.)

c. Prove that median [pic] is also altitude of the triangle.

2. Triangle JKL has vertices at J(6, 0), K(10, 6) and L(0, 2).

a. Find the coordinates of M and N, the midpoints of [pic] and [pic] respectively.

b. Prove that [pic] is parallel to [pic].

c Prove that [pic] is half the length of [pic].

3. Triangle PQR has vertices P(7, 1), Q(–1, 3) and R(2, 6).

a. Prove that ΔPQR is a right triangle.

b. Prove that the length of the median from the right angle to the hypotenuse is half the length of the hypotenuse. (Note: This turns out to be true for all right triangles.)

Geometry HW: CG - Review

Name

If you want graph paper, it’s at the mailboxes. Help yourself (within reason).

1. What is the slope of the line containing points A(a, b) and B(a – 4, b + 2c)?

2. Find the coordinates of the midpoint of the segment whose endpoints are (a, b) and (–5a, 7b).

3. If M(1, 2) is the midpoint of [pic] and the coordinates of A are

(3, –1), find the coordinates of B.

4. Given points A(–2, 2), B(3, 7), C(5, –1), and D(k, 2). If [pic], find the value of k.

5. Find the distance between the points (a, b) and (a + 2b, 4b).

6. Which is an equation of the set of points that are 4 units from the point (–3, 2)? (Circle one.)

(1) (x + 3)2 + (y – 2)2 = 2 (2) (x + 3)2 + (y – 2)2 = 16

(3) (x – 3)2 + (y + 2)2 = 2 (4) (x – 3)2 + (y + 2)2 = 16

  7. What is the radius of a circle whose center is at (–2, 6) and passes through the point (0, 3)?

 8. Draw the graphs x2 + y2 = 4 and y = 4 on the same axes. How many points are common to both graphs?

  9. What are the coordinates of the center and the length of the radius of the circle [pic]?

10. Triangle ABC has vertices A(–4, 7), B(6, –3) and C(2, 9).

a. Prove using coordinate geometry that (ABC is a right triangle.

b. Prove using coordinate geometry that the length of the median from C to [pic] is half the length of [pic].

11. The vertices of ΔABC are A(2, 5), B(4, –1), and C(–3, 0).

a. Prove that ΔABC is isosceles.

b. Find the coordinates of M, the midpoint of [pic].

c. Find the length of the median from C to [pic].

d. Prove that [pic] ⊥ [pic].

12. Given: The vertices of quadrilateral ABCD are

A(0, 0), B(s, 0), C(t + s, s), and D(t, s). If s > 0

and t > 0, prove using coordinate geometry that:

a. ABCD is a parallelogram. (A parallelogram

is a quadrilateral with both pairs of opposite

sides parallel.)

b. ABCD is not a rhombus. (A rhombus is a quadrilateral with all four sides congruent.)

13. Air pressure above the Earth’s surface can be approximately modeled by p = –10.36a + 100.56 where p is the pressure in kilopascals (kPa) and a is the altitude above the Earth’s surface in kilometers. Based on this equation,

a. what is the air pressure at the Earth’s surface? (Think: how high is the Earth’s surface above the Earth’s surface?)

b. what happens to air pressure each time you move one kilometer higher above the Earth’s surface?

Stuff you should know:

Vocabulary

slope, midpoint, distance

parallel, perpendicular

congruent

bisect (bisector)

collinear

median, altitude, angle bisector

axis of symmetry

y-intercept

vertex

Formulas and equations

Slope formula

Distance formula

Midpoint formula

Equations of lines

Horizontal line

Vertical line

Slope-intercept form

Equation of a circle

How to:

Graph a line

Graph a linear inequality

Graph a circle

Determine if two segments

are congruent

are parallel

are perpendicular

bisect each other

Write the equation of a line given two points

Write the equation of a circle given its center and radius

Find the center and radius of a circle given its equation

Completer the square

Find the point that divides a segment in a given ratio

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(s, 0)

(s + t, s)

(t, s)

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