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Concepts of Geometry

Homework Packet
Page 1
Medians, Altitudes and Angle Bisectors

Part One. Draw each of the following. Make sure all parts are labeled and marked
appropriately.

1. PS is a median of PQR and                           2. QT is an angle bisector of PQR
S is between Q and R                                    and T is between P and R

3. AC and BC are both altitudes of ABC                 4. PT and RS are medians of PQR
and intersect at point V

Part Two. Tell if each of the following statements are ALWAYS, NEVER, or
SOMETIMERS true.

5. The three medians of a triangle intersect at a point inside the triangle.

6. The three angle bisectors of a triangle intersect at a point outside the triangle.

7. The three altitudes of a triangle intersect at a vertex of the triangle.

8. If a median of a triangle is also an altitude, then the triangle is isosceles

9. A point on the angle bisector of a triangle is equal distances from the sides of the
triangle.
Page 2
Page 3

For questions #1-4, use a triangle with the coordinates of R(3, 3) S(-1, 6) and T(1, 8)
1. If RX is a median of this triangle, what are the coordinates of X?

2. What is the length of segment RX?

3. What is the slope of RX?

4. If SP is an altitude of the triangle, what is the slope of SP?

Use this information for questions #5-7: YV is an angle bisector of XYZ and
ZYX = 8x – 6 and XYV = 2x + 7.
5. What is the value of x?

6. What is the measure of ZYX?

7. What is the measure of ZYV?

8. Find the values of x and y if
AD is an altitude and a median
of the triangle.

.

4x - 6
10y - 7            5y + 3
Page 4
Solving Triangles
Solve for the value of x and y in each figure if ABC  DEF

1. C                                          F
4y
28
18
E              D
2x + 12
A                       B
20

2.                   B

44                                                  2x - 20
40                                           D
E

A                          C                  4y + 12
36

F

A
3.
E

65                     45
5x + 35
10y

D
F
50

B
C

Page 5
Triangle Sides and Angles
Part One. For #1-2, list the angles of each triangle in order from greatest to least.

1.                                             2.          M
A
7.2

11                                                                      L
9                            8

7.8
C                            B
8
N

Part Two. For #3-4, list the sides in order from shortest to longest.

3.                                                    4.
R
F
45
30                                         Q 80

55
89
61
H                                    G                             S

Part Three. For #5-6, use the given information about a PQR to place the sides in order
from longest to shortest.
5. P = 7x + 8, Q = 8x – 10 and R = 7x + 6

6. P = 3x + 44, Q = 68 – 3x and R = x + 61
Page 6
Triangle Inequalities
Part One. Determine whether each set of numbers could be the sides of a triangle. Write
YES or NO.

1. 12, 11, 17                       2. 1, 2, 3                    3. 4, 9, 4

4. 2.5, 6, 6.5                      5. 12, 2, 14                  6. 2, 12, 12

7. 9, 40, 41                        8. 5, 100, 101                9. 204, 7, 215

Part Two. Two sides of a triangle are 18 and 21. Tell if each of the following numbers
could be the third side of that triangle. Write YES or NO.

10. 10                              11. 40                        12. 7

13. 21                              14. 3                         15. 57

Part Three. The measures of two sides of a triangle are given. Between what two
numbers must the third side be?

16. 12 and 15

17. 4 and 13

18. 21 and 17

Page 7
Page 8
GUESS AND CHECK (Part one)

1. Sum of two numbers is 15
Product of those numbers is 56.
Find the two numbers.

2. Sum of two numbers is thirteen.
Difference of those numbers is 3.
Find those two numbers.

3. Sum of two numbers is 18.
Quotient of those numbers is 2.
Find those two numbers.

4. Sum of two numbers is 32.
Product of those numbers is 231.
Find those two numbers

5. Sum of three numbers is 17.
Product of those numbers is 168.
Find all three numbers.
Page 9
GUESS AND CHECK (Part two)

1. On a farm, there are chickens and rabbits. All together, there are seven
heads, but 20 legs. How many of each type of animal was on the farm?

2. A teacher tells his class that their next test is worth 100 points and has 30
problems on it. Some of the problems are open ended which are worth 6
points while others are multiple choice and only worth two points. How
many of each type of question are on this test?

3. John has quarters and dimes in his pocket. When counted together, he
found he has \$1.05 with just six coins. How many of each coin does he have
in his pocket?

4. Tracy’s three math test grades had an average of 80. Her first test was
ten points more than her second. The second was test was twenty points less
that the third. What were her three test grades?

5. Find three consecutive odd numbers whose sum is 39.
Page 10
TOO MUCH INFORMATION

1. Gracy High School ordered 17 computers and 9 dvd players. The
computers sold for \$675 each, but the school got a discount and only paid
\$558 each. Each dvd player cost \$65 each. How much did the school pay
total for the computers?

2. Harry works 17 hours a week and makes \$9.50 an hour. He also spends
10 hours a week practicing his trumpet. He saved all his earning for seven
weeks and purchased a new trumpet for \$675. After this purchase, how
much of his savings did Harry have left?

3. The fastest speed that Mona can get her car to go is 80 miles per hour.
However, Candy can get her car to 92 miles per hour and Janice gets her car
up to 98 miles per hour. If these women drive their cars at their fastest rates
possible, then how much further does Janice travel in a half hour than Mona
does?

4. Lily bought 6 hamburgers at 79 cents each, nine French fries at 72 cents
each, six sodas at \$1.09 each, and two milkshakes for \$2.15 each. How
much did Lily spend total on the drinks?

Page 11
WORKING BACKWARDS

1. Susan made a deposit of \$47 into her checking account. Then, she wrote
a check for \$112. After that, her balance was \$1102. How much was in her
account originally before the deposit and check cashing?

2. Ace gave his friend Benny 12 of his marbles. After that, he had 16
marbles left. How many marbles did Ace originally have?

3. Begin with a number. Add seven to that number. Multiply that result by
three. Then, subtract ten from that result. The final number is 62. Find the
original number.

4. Gary told his niece that is you double his age and then add three, you get
87. How old is Gary?

5. A man got into an elevator on a specific floor. He rose two floors, then
descended three floors, then rose up again four floors. If he then got off on
the 7th floor, on what floor did he begin?

Page 12
MAKE A LIST

1. A DJ has four different songs to play. He hasn’t decided the order to play
them in yet, though. How many different choices does he have?

2. A vending machine will take nickels, dimes or quarters. Each item in the
machine costs 65 cents. How many different combinations of coins could be
used to purchase an item?

3. A student must answer any four of five questions on a history exam.
How many different choices does that student have to answer four of five
questions?

4. Sarah bought 2 skirts, four blouses, and 3 pairs of shoes. How many
different combinations of outfits does that give her?

Page 13
MAKING A TABLE

1. Joey placed \$400 into a savings account. Each year, 5% interest is
gained on the amount in the account. Complete this table if Joey does not
withdrawal any money:
Year          0             1            2           3             4
Amount         \$400

2. Mitchell wrote a book and had it published. The first month it was
released, he sold 5 copies. The second month, he sold 15 copies. In the
third month, there were 35 sales and in the fourth, there were 65. Complete
this table to continue such a pattern:
Month            4th          5th          6th           7th        8th
Copies           65

3. Zeus had \$600 in his savings account, but withdrew \$25 each month.
Kara started with \$340, but deposited \$70 each month. Complete this table.
Month           0            1            2           3            4
Zeus \$        \$600
Kara \$        \$340

4. Betty sells baked goods at a local market stand. She currently charges
\$1.20; however, she wants to raise the price to \$2.10. She does not want to
do it all at once, though. She will raise the price by the same amount each
week until the price reaches \$2.10. Complete this table based upon such
information:
Week           0          1           2           3          4         5
Price        \$1.20                                                   \$2.10

Page 14
DRAW A PICTURE

1. Rose has a garden which is 10 feet long and six feet wide. She wants to
place a fence around her garden. To do so, she first places posts in the
ground. She begins in one corner, and places a post every 2 feet apart. How
many posts will she need?

2. Amber flew a kite the highest of all her friends. Bonnie’s kite was 50
feet lower than Amber’s but 100 feet higher than Carla’s. Carla’s kite was
100 feet higher than Derek’s. Derek’s kite was 120 feet above the ground.
How high was Amber’s kite?

3. A round circle has three straight line cuts made into it. What is the
largest number of pieces that it could be cut into?

4. The deck in Mary’s yard is in the shape of a circle. The circle has 6 posts
dispersed around its circumference. If you want to attach one string from
each post to every other post, what is the least number of strings that will be
needed?

5. Gary started at a point and walked 20 feet north. He then went 40 feet
east, 20 feet south, and 10 feet west. How far is Gary from his starting
point?

Page 15
Page 16
USING THE HSPA REFERENCE SHEET

1. How many ounces are in three pounds?

2. What fraction is equal to pi (  )?

3. Find the volume of a rectangular prism when its length is 3, width is 5 and height is 6.

4. Find the surface area of the same prism from question #3.

5. How many seconds are in one day?

6. Which is longer: four feet, one yard OR forty inches?

7. How many degrees are in a circle?

8. Find the area of a circle when its radius is 5.

9. How many fluid ounces are in one quart?

10. A rectangle has a perimeter of 40 and a width of 6. Find its length.

Page 17
HSPA Multiple Choice questions
1. Hank receives an allowance each week from his parents. He is required to save 10%
of it in a bank account. What would Hank’s minimum allowance be so that he would
have \$15 to actually spend?
A. \$13.50            B. \$15.10           C. \$16.50             D. \$16.67

2. In the five basketball games of the Holiday Tournament, four of the scores for Joe’s
team were 109, 105, 97 and 92. If the mode of all five scores was 92, what was the mean
of those five scores?
A. 99                 B. 97                 C. 95                 D. 93

3. Jason’s older brother owes him \$200. He offers to pay Jason \$120 today and then
tomorrow to pay him ¼ of what he paid him today. If this pattern of paying ¼ of what he
paid the day before continues, how much will Jason’s brother have paid by the end of the
third day?
A. \$160.00            B. \$157.50           C. \$155.00            D. \$149.75

4. Jan is building a scale model of a house. If the actual house is 86 feet wide and 172
feet long, what will be the length in inches of the model if it is 18 inches wide?
A. 3                   B. 6                   C. 12                   D. 36

5. Alex jogged 8 miles north, then turned due west and jogged 6 more miles. What is the
shortest distance from Alex’s current place to his starting point?
A. 9                  B. 10                   C. 14             D. 24

6. Laura wanted to raise her science grade from 80 to 84. What percent increase would
that represent?
A. 4%                 B. 5%                 C. 6%                D. 7%

7. The Garden Center Manager, Mr. Stern, ordered two sizes of grass seed bags for the
spring sale. He ordered three times as many of the 25-pound bags as he did of the 40-
pound bags. The total weight of all of the seed was 920 pounds. How many smaller bags
of seed did Mr. Stern order?
A. 6                  B. 18                 C. 24                D. 81

Page 18
Part Eight:

Page 19
Parallelograms

Part One. Use parallelogram ABCD to complete the blanks in each
statement.

1. AB is parallel to _____
A                        B
2. DA  _____

4. CDA  _____
D                            C

5. DE is congruent to _____

6. BAC  _____

Part Two. Use parallelogram EFGH to tell if each statement is true or false.

7. FE || GH

8. FDE  HDG                                F

9. FGH  FEH                                                                     G

10. FD is congruent to DG
D

11. FHE  GHE                           E

12. DE = ½EG                                                                   H

Page 20

1. If quadrilateral SLAM is a parallelogram and S = 92, then find the
measures of the other three angles.

2. If DUNK is a parallelogram, DU = 3x + 6, UN = 8y – 4, KN = 8x – 4 and
KD = 2y + 14, then find the perimeter of DUNK.

3. Given parallelogram PQRS with P = y and Q = 4y + 20, find the
measure of R

4. In parallelogram ABCD, if A = x + 75 and B = 3x – 199, then find
the measure of D.

5. Find the values of x and y:
6y

30                     5x + 10

For #6-8, draw a parallelogram ABCD with diagonals intersecting at point
T. Use that figure to answer these questions.

6. If ABC = 137, then find DAB

7. If AC = 5x – 12 and AT = 14, then find the value of x.

8. If BC = 4x + 7 and AD = 8x – 5, then find the value of x.
Page 21
Rectangles

Use rectangle JKLM to answer questions #1-2 to find the value of x:
M
1. LP = 3x + 7 and MK = 26                       L

2. LJ = 4x – 12 and KM = 7x – 36                        P

K                    J

Use rectangle MATH to answer questions #3-8:

3. If MP = 6, then find HA

4. If MH = 8, then find AT                   M                            A

P
5. If HP = 3x and PT = 18, then find x

H                                    T
6. If MPH = 55, then find MHP

7. If MPA = 110, then find APT

8. If MA = 2x + 4, AT = 8, TH = 3y + 5, and MH = 2y – 2, then find the
values of x and y.

Page 22
Squares

For #1-5, draw a square ABCD with diagonals intersecting at point E. Use
that figure to answer the questions.

1. If AEB = 3x, then find the value of x.

2. If BAC = 9x, then find the value of x

3. If AB = 2x + 4 and CD = 3x – 5, then find the length of BC

4. If DAC = y and BAC = 3x, then find the values of x and y.

5. If AB = x + 15 and BC = 2x, then find the length of CD.

6. If WXYZ is a square, then what is the measure of ZXY?

Page 23
Rhombus

#1-6: Use rhombus BEAC and the fact that BA has a length of 26 to tell if
each statement is TRUE or FALSE.

1. CE = 26

2. HA = 13                                             B                       E

3. BA  EC                                                             H

4. BHE  AHC
A
C
5. BEH  EBH

6. CBE and BCA are supplementary.

For #7-10, use rhombus SRQP.

7. If ST = 13, then find SQ

S                       R
8. If PRS = 17, then find QRS

T

9. What is the measure of STR?                            P                       Q

10. If SP = 4x – 3 and PQ = 18 + x, then find the value of x.

Page 24

Complete each box with a “YES” or “NO”.

Property      Parallelogram    Rectangle    Square   Rhombus    Trapezoid
Opposite
sides are
equal

Diagonals
are
perpendicular

All four sides
are equal

Opposite
sides are
parallel

Diagonals
bisect each
other

Diagonals
are congruent

Only one pair
of parallel
sides

Diagonals
bisect
opposite
angles
Page 25
Trapezoid

For #1-2, solve for x. (Assume the line inside the trapezoid is a median)
3x + 5
1.           27                        2.

x                                              55

13

7x - 10

For #3-6, use trapezoid PQRS as seen to the right. TV is its median.

3. IF PS = 20 and QR = 14, then find
the length of TV

P
T
4. If QR = 14.3 and TV = 23.3, then                                            Q
find the length of PS

5. If TV = x + 7 and PS + QR = 5x + 2,                                             R
then find the value of x                                             V
S

6. If RVT = 57, then find QTV.

7. UR is the median of a trapezoid with bases ON and TS. If the
coordinates of the points are U(1, 4) R(8, 4) O(0, 0) and N(8, 0), then find
the coordinates of points T and S.
Page 26
Page 27
Proportions

Tell if each set of fractions forms a proportion. Write “YES” or “NO”

6 22                4 16                   4 12            8 16
1.                  2.                   3.             4.     
8 28                5 20                  11 33            9 17

Solve each proportion for the value of x. (Note: some answers may be
decimals)

3 x                                x   3                   x 7
5.                               6.                      7.     
4 8                               45 15                    9 16

x .21                             858 702                   x 11.75
8.                               9.                      10.     
8 2                                x   900                 33 35.25

x 3                               3 1                     3 12
11.                              12.                     13.     
4 2                               x 6                     1   x

Substitution. If a = 3, b = 2, c = 6, and d = 4 , then do each of the following
form a proportion? Write “YES” or “NO”

b d                               a b                     c d
14.                              15.                     16.    
a c                               c d                     b a

ab cd                           d c                     ab cd
17.                              18.                     19.      
b   d                            b a                      c   b

Page 28

Solve each proportion for the value of x. Some answers will be decimals.

3 x2                                    x  2 14
1.                                      2.         
5   6                                      5    10

5  x 14                                 x  9 x  10
3.                                      4.         
x  3 10                                  10     11

x  7 x  12                             x2 5
5.                                      6.       
4     6                                 7   7

x  1 1                                 1   2
7.                                      8.      
3    5                                 3 x2

x  2 x 1                               5  x 13
9.                                      10.        
3     6                                3  x 12

Page 29
Proportion Word Problems

Solve each word problem by using proportions.

1. A 96-mile trip requires 6 gallons of gas. At that rate, how many gallons would be
needed for a 152 mile trip?

2. When a pair of blue jeans is made, the leftover denim scraps can be recycled to make
pencils. One pound of denim is left over after making every five pairs of jeans. How
many pounds are left over after 250 jeans are made?

3. In the first 30 minutes of the opening day of the Texas State Fair, 1252 people entered
through the gates. If this rate continued, how many people would enter during the
operating hours of 8am to 11pm?

4. Josh finished 24 math problems in one hour. At that rate, how many hours will it take
him to complete 72 problems?

5. To make a model of the Guadelupe River bed, Harold used 1 inch of clay for every 5
miles of actual river length. His model was 50 inches long. How long is the actual river?

6. A recipe calls for 5 pounds of cocoa beans for every 4 gallons of milk. How much
milk is needed if 75 pounds of cocoa beans is used?

Page 30
Similar Figures
(Figures in this lesson are not drawn to scale)
Use the similar quadrilaterals to the right for questions #1-3:
(Note: QRSP  BCDA)                                                           Q                             R

1. If AB = 4, AD = 8 and PQ = 6, find PS

P                             S

2. IF RS = 4.5, CD = 6.3 and BC = 7, find QR.
B                     C

3. If CD = 22, DA = x - 2 and SR = 33,                                    A
find SP.                                                                                      D

Use the trapezoids to the right to answer questions #4-8. Note that TFEL  KROC.

4. How long is KR?                               15
T                             F

12                       12
5. How long is RO?

K                         R
6. How long is CO?                           L           E
6                            8

C             O
7. What is the perimeter of TFEL?

8. What is the perimeter of KROC?

Use the quadrilaterals to the right to answer questions #9-11. Note that BCDA  QRSP.
9. What is the value of x?              B       x+5
Q x+3
5                            C                                          R
10. How long is BC?                                                                   4

A                                                     P                 S
11. How long is QR?
D                                        Page 31
Figure Proportions
For #1-4, tell if each proportion is TRUE or FALSE.
C
BC AB                  CB CA
1.                    2.   
ED CD                  CD CE
B                   D
AB DE                  BA CA
3.                    4.      
BC DC                  DE CE
E
A

For #5-10, fill in the blanks with the correct segment.
D
YO AE                  YB   ?
5.                    6.                                                                A
OB   ?                 OB ER                                    Y

?   YB                DY DA                                                        E
7.                    8.       
AE YO                  YO   ?                               O
R
DR DB                    ?   DO
9.                    10.    
?   YB                 AE YO                          B

For #11-14, tell if each proportion is TRUE or FALSE.
T           L
A
TM TL                  BA ML
11.                   12.   
MB LA                  MB LA

TB TA                  TM TB                                            M
13.                   14.     
MB LA                  ML BA

B

Page 32

Solve for all missing values (x and y)

8              6
1.        7                      x                2.
x+5                  x
10            12
2y + 6                       2x

x + 12
3.                                          4.
y+8
5x - 3                                      9

2y - 4                                                       x
24

18

5.                                                6.
4
5
6                                                            x-3

x                                                     7
x+3
x+3

Page 33
REVIEW

1. Use a triangle with the coordinates of R(3, 5) S(-1, 10) and T(1, 4)
(a) If RX is a median of this triangle, what are the coordinates of X?

(b) What is the length of segment RX?

(c) What is the slope of RX?

(d) If SP is an altitude of the triangle, what is the slope of SP?

2. Solve for the value of x and y in these figures if ABC  DEF

C                                  F
4y
36
16
E             D
2x + 12
A                 B
20

3. Use the given information about a PQR to place the sides in order from longest to
shortest.
P = 2x + 8, Q = 5x – 5 and R = 3x + 7

4. Two sides of a triangle are 15 and 21. Tell if each of the following numbers could be
the third side of that triangle. Write YES or NO.

(a) 36                                 (b) 40                        (c) 7

5. The measures of two sides of a triangle are given. Between what two numbers must
the third side be?
12 and 19

Page 34
6. Solve each word problem:
(a) Sum of two numbers is 18 and the product of those numbers is 80. Find the two
numbers.

(b) On a farm, there are chickens and rabbits. All together, there are nine heads, but 38
legs. How many of each type of animal was on the farm?

(c) Begin with a number. Add seven to that number. Multiply that result by two. Then,
subtract ten from that result. The final number is 24. Find the original number.

(d) A vending machine will take nickels, dimes or quarters. Each item in the machine
costs 75 cents. How many different combinations of coins could be used to purchase an
item?

(e) Zeus had \$800 in his savings account, but withdrew \$25 each month. Kara started
with \$450, but deposited \$60 each month. Complete this table.
Month             0             1            2                   3              4
Zeus \$          \$800
Kara \$          \$450

(f) Gary started at a point and walked 25 feet north. He then went 47 feet east, 25 feet
south, and 10 feet west. How far is Gary from his starting point?

(g) How many seconds are in one day?

(h) Jason’s older brother owes him \$260. He offers to pay Jason \$120 today and then
tomorrow to pay him ¼ of what he paid him today. If this pattern of paying ¼ of what he
paid the day before continues, how much will Jason’s brother have paid by the end of the
third day?

Page 35
7. If quadrilateral SLAM is a parallelogram and S = 91, then find the measures of the
other three angles

8. Use rectangle JKLM to answer questions to find the value of x:
(a) LP = 3x + 7 and MK = 26
M
L
P

(b) LJ = 4x – 12 and KM = 6x – 36
K                               J

9. Draw a square ABCD with diagonals intersecting at point E. Use that figure to
(a) If AEB = 5x, then find the value of x.

(b) If BAC = 9x, then find the value of x

(c) If AB = 2x + 4 and CD = 3x – 9, then find the length of BC

10. For #3-6, use trapezoid PQRS as seen to the right. TV is its median.
(a) IF PS = 20 and QR = 16, then find
the length of TV

P
(b) If QR = 14.3 and TV = 25.3, then                                            T
Q
find the length of PS

(c) If TV = x + 7 and PS + QR = 5x + 2,                                                      R
then find the value of x
V
S

(d) If RVT = 59, then find QTV.
Page 36
For #11-14, solve each proportion.
4 x2                                           x  2 14
11.                                           12.        
5   10                                            5    10

x5 7                                          x  8 x  10
13.                                           14.        
x 3 5                                           5     4

15. A 96-mile trip requires 8 gallons of gas. At that rate, how many gallons would be
needed for a 152 mile trip?

16. Use the similar quadrilaterals to the right for questions :
(Note: QRSP  BCDA)                                                          Q                 R

(a) If AB = 4, AD = 5 and PQ = 8, find PS

P                      S

(b) IF RS = 4, CD = 3 and BC = 6, find QR.
B             C

(c) If CD = 22, DA = 32 and SR = 33,                                     A
find SP.                                                                             D

17-18: Solve for x

.17                                            18.
4
6
8                                                           x-3

x                                              8
x+3
x+4

Page 37

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