Final Exam Review Packet Geometry Regents Review

Final Review by topic worksheets

Final Review Mixed worksheets

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A B

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Final Review #1 (by topic) Name ______________________________

Geometry Date _______________ Block ________

Exterior angles of a polygon

1) Two angles of a triangle have measures of 80 and 40. Which is not the measure of

an exterior angle of the triangle?

(a) 1200 (b) 1000 (c) 1100 (d) 1400

2) If the measure of an exterior angle of a regular polygon is 720, then the polygon is:

(a) a decagon (b) an octagon (c) a pentagon (d) a square

3) The number of degrees in the measure of one exterior angle of a square is:

(a) 600 (b) 1800 (c) 2700 (d) 900

______________________________________________________________________

Similarity and proportions

4) The sides of a triangle have lengths 3, 5, and 7. In a similar triangle, the shortest side

has length x-3, and the longest side has length x+5. Find the value of x.

5) In the diagram, ~CDE CAB . If CD = 8,

CE = 6, and EB = 5, find AD.

6) In the diagram, ~ADE ABC . Given that AD = 4,

DB = 3, and EC = 4.5, find AE.

________________________________________________________________________

Complementary and supplementary angles

7) Two complementary angles are in the ratio of 7:2. Find the number of degrees in

the smaller angle.

8) If two angles are supplementary and the measure of <A is 12 less than twice the

measure of <B, find the larger of the two angles.

9) Angles A and B are complementary. If the measure of <B is 2 greater than three

times the measure of <A, find the smaller of the two angles.

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Ratio of areas

10) The ratio of the radii in two circles is 3:7. What is the ratio of the area of the smaller

circle to the larger circle?

11) The ratio of the corresponding sides of two similar polygons is 1:4. Find the ratio of

their areas.

12) If the ratio of the corresponding sides of two similar polygons is 2:3, and the area of

the larger triangle is 243, find the area of the smaller triangle.

_______________________________________________________________________

Geometric probability

13) Find the probability that a penny tossed at

random onto the figure will land in the shaded

region. The length of a side of the square is 4 cm.

Round to the nearest hundredth.

14) Find the probability that a dart tossed at

random onto the figure will land in the shaded

region. The radius of the circle is 2.

Round to the nearest hundredth.

15) Find the probability that a dart tossed at

random onto the figure will land in the

shaded region.

________________________________________________________________________

Segment addition

16) Given that A, B, and C are collinear with A-C-B, if AB = 25, AC = 3x+1, and CB = x,

find the value of x.

17) Given P, Q, and R are collinear with P-Q-R, with PQ = 2x, QR = 3x, and

PR = 25. Find the value of x.

18) If E, D, and F are collinear with E-D-F, ED = 5x, DF = 3x, and EF = 25, find x.

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Final Review #2 (by topic) Name _____________________

Geometry Date ________ Block _____

Quadratic functions

Write the coordinates of the vertex and state the equation of the axis of symmetry for

each parabola.

1) 2 4 3y x x 2) 2 2 8y x x 3) 2 2 3y x x

______________________________________________________________________

Parallelograms

4) In parallelogram ABCD, AB = 5 4x and CD = 2 14x . Find the value

of x.

5) In parallelogram ABCD, m<A = 2x, and m<B = 2x+20. Find the value

of x.

6) If the degree measures of two consecutive angles of a parallelogram are

represented by x+40 and 2x-10, find the value of x.

______________________________________________________________________

Pythagorean theorem

7) If the hypotenuse of a right triangle is 10 and one leg is 6, find the length of the

other leg.

(a) 64 (b) 16 (c) 8 (d) 4

8) In an isosceles right triangle, one leg is 3. Find the length of the hypotenuse.

(a) 3 2 (b) 6 (c) 3 (d) 3

9) In a right triangle, one leg has a length of 3 and the hypotenuse has a length of 10.

What is the length of the other leg?

(a) 91 (b) 7 (c) 109 (d) 91

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Indirect proof

10) To prove indirectly that AB CD , what assumption must be made?

11) To prove indirectly that BD is not the perpendicular bisector of AC , what

assumption must be made?

12) To prove indirectly that ABC EFG , what assumption must be made?

_______________________________________________________________________

Surface area

13) If the surface area of a cube is 96 cubic centimeters, what is the length of a side

of the cube?

(a) 3 cm (b) 4 cm (c) 5 cm (d) 6 cm

14) A cereal box is a rectangular prism 30 cm high. The sides of the base measure 8

cm and 25 cm. Find the surface area of the box.

15) What is the surface area of a cube with a side length of 4?

______________________________________________________________________

Midpoint

16) Find the coordinates of the midpoint of the segment whose endpoints are (-2, 3)

and (4, -3).

17) Line segment AB has midpoint M. If the coordinates of A are (2, 3) and the

coordinates of M are (-1, 0), what are the coordinates of B?

18) Find the midpoint of the line segment formed by the points (5, 4) and (-3, -4).

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Final Review #3 (by topic) Name ___________________

Geometry Date _________ Block _______

Triangle inequality

1) In ABC , AB = 14, and BC = 9. AC may not be equal to:

(a) 5 (b) 13 (c) 23 (d) 25

2) If the lengths of 2 sides of a triangle are 4 and 8, the length of the third side may

NOT be:

(a) 5 (b) 6 (c) 7 (d) 4

3) Which of the following sets may represent the lengths of the sides of a triangle?

(a) {2, 4, 6} (b) {4, 7, 12} (c) {7, 12, 5} (d) {8, 10, 14}

______________________________________________________________________

Transformations

4) Find the coordinates of the image of point T(-7, 3) under a reflection in the origin.

5) What are the coordinates of R’, the image of R(-4, 3) after a reflection in the x-axis?

6) What are the coordinates of N’, the image of N(5, -3) under a translation such that

, 3, 4x y x y ?

_______________________________________________________________________

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Equations of circles

7) What are the coordinates of the center of a circle represented by the equation

2 2

2 3 49x y ?

8) State the equation of a circle which has a radius of 5 and has a center with the

coordinates (-2, 6).

9) A circle whose center is a point (1, 2) passes through a point (4, -2). What is the

length of the radius?

________________________________________________________________________

Midsegments

10) A triangle has sides of 3, 5, and 10. What is the perimeter of a triangle formed by

connecting the midpoints of the sides of the triangle?

11) If a triangle has sides of 15, 20, and 25, which of the following could be the length

of a midsegment of the triangle?

(a) 15 (b) 10 (c) 9 (d) 12

12) In a triangle, which of the following is always parallel to a side of the triangle?

(a) median (b) altitude (c) midsegment (d) hypotenuse

________________________________________________________________________

Rational Equations

Solve the following equations.

13) 2 3 1

4 3 2

r r

14) 5 3

132x x

15) 3 1

12

x

x x

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Final Review #4 (by topic) Name ______________________

Geometry Date ______________ Block _____

Triangle congruence

1) In the accompanying diagram, RL LP ,

LR RT , and M is the midpoint of TP . Which

method could be used to prove TMR PML ?

(a) SAS (b) AAS (c) HL (d) SSS

2) In the accompanying diagram, ,ACE BCD ,

A E and C is the midpoint of AE . Which

theorem justifies ABC EDC ?

(a) SSS (b) SAS (c) ASA (d) SSA

3) In the diagram of isosceles triangle ABC,

<ACB is the vertex angle, CM AB , and M is

the midpoint of AB . Which statement can not

be used to justify ACM BCM ?

(a) HL (b) AAS (c) SSS (d) AAA

_____________________________________________________________________

Special triangles

4) If the shortest side of a 30-60-90 triangle has length x, then the hypotenuse has

length:

(a) x (b) 2x (c) 3x (d) 2x

5) If the leg of a 45-45-90 triangle has length z, then the hypotenuse has length:

(a) z (b) 2z (c) 3z (d) 2z

6) If a rhombus has a side length of 2 and a 60 degree angle, what are the lengths of

the diagonals of the rhombus?

(a) 2 and 2 3 (b) 1 and 3 (c) 2 and 2 2 (d) 1 and 2

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Final Review #5 (mixed) Name ______________________

Geometry Date _________ Block _________

1) An equilateral triangle has a side length of 20. What is the length of the altitude?

(a) 40 (b) 10 3 (c) 20 2 (d) 20

2) In Triangle ABC, if AB < BC < AC, then which of the following statements is false?

(a) m A m C (b) m A m B (c) m B m C (d) m B m A

3) Which of the following statements is always true?

(a) The diagonals of a parallelogram are congruent.

(b) The diagonals of a parallelogram bisect the angles of the parallelogram.

(c) The diagonals of a parallelogram bisect each other.

(d) The diagonals of a parallelogram are perpendicular to each other.

4) The degree measures of two supplementary angles are 3x-17 and 5x+21. Find the

measures of both angles.

5) Find the probability that a dart tossed at random onto the figure will land in the

shaded region. State the answer as a fraction in simplest form.

6) The diagonals of two rectangles which are similar measure 5 and 15 respectively. If

the area of the smaller rectangle is 27, find the area of the larger rectangle.

7) Which whole number, when substituted for x, makes the following statement true? 3 6 4x and x

8) In parallelogram LMNO, an exterior angle at vertex O measures 72. Find the

measure of angle L of the parallelogram.

9) To prove indirectly that YT is a median, what assumption must be made?

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10) Given the equation 22 8y x , find the vertex and the axis of symmetry.

11) Given line segment AB with C between A and B, if AC = x+1, CB = 2x, and AB = 19,

find the value of x.

12) The length of the hypotenuse of a right triangle is 17 meters and the length of one

leg is 15 meters. What is the length of the other leg?

13) What is the length of a side of a cube with a surface area of 150?

14) If the midpoint of a line segment is (1.5, -1) and one of the endpoints is (-2, 7), find

the other endpoint of the line segment.

15) A translation maps A(2, 5) onto A’(-3, 7). What are the coordinates of the point (-

3, 0) under the same translation?

16) What is the radius of a circle whose center is at the origin and that passes through

the point (4, 0)?

17) A triangle has sides of 7, 10, and 20. What is the perimeter of a triangle formed by

connecting the midpoints of the sides of the triangle?

18) Given XYZ STU , name all congruent sides and all congruent angles.

19) In a right triangle ABC, CD is drawn perpendicular to hypotenuse AB . If AB = 16,

and DB = 4, find BC.

10

70o

60o

x

x160 o

5

4

x

6

88 o

120 o

x

40 o

x16

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Final Review #6 (mixed) Name _____________________

Geometry Date __________ Block _____

Find the value of x.

1) 2) 3)

4) 5) 6)

______________________________________________________________________

7) Given a circle with a radius of 5 and a center of (2, -3), write the equation of the

circle.

8) Given circle O with A and B on the circle and m<AOB = 80, find the length of arc

AB (round to the nearest tenth if necessary).

9) Given: TU is tangent to Circle P at point T; 90mQR , 150mRT , and 50mQS .

Find m STU , 1m , and 2m .

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10) The vertices of quadrilateral ABCD are A(1, 1), B(3, 4), C(9, 1), and

D(7, -2).

a) Prove that ABCD is a parallelogram.

b) Prove that ABCD is not a rectangle.

11) Quadrilateral DEFG has vertices D(-4, 0), E(0, 1), F(4, -1), and G(-4, -3).

a) Prove that DEFG is a trapezoid.

b) Prove that DEFG is or is not an isosceles trapezoid.

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Final Review #7 (mixed) Name _____________________

Geometry Date __________ Block _____

1. What is the equation of the circle with a radius of 7 and a center of

(3, -5)?

2. The larger of two supplementary angles is 40 more than three times the smaller

angle. Find the measure of the larger angle.

3. If a triangle has sides of 30, 15, and 41, what is the perimeter of the triangle formed

by connecting the midpoints of the sides of the triangle?

4. What is the image of (5, -1) after a reflection in the y-axis?

5. Midpoint M of segment AB has coordinates (4, -3). If the coordinates of A are (2, 0),

what are the coordinates of B?

6. In parallelogram MATH, m<T = x+20 and m<H = 2x+10. Find the value for m<A.

7. In the diagram, ~ABC AEG . If AE = 10,

EB = 4, and GC = 5, find the value of AG.

8. In triangle TAP, m<T = 50, m<A = 100. Which side of the triangle is the smallest?

9. A square has a side of 16. What is the length of a diagonal of the square?

10. Find the probability that a dart tossed at random will land in the shaded area of

the figure.

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Mixed Review 1:

1. Graph: 2 2( 1) ( 3) 9x y 2. Write a statement that is

logically equivalent to ”If an angles

is a straight angle, then its measure

is 180 degrees”? _________________________________

_________________________________

_________________________________

__________________________

__________________________

3. Given parallelogram ABCD. If 2 14m A x and 10 50m B x , find the positive

value of x.

4. Intersecting lines a and b are in plane R. Line m is perpendicular to both

lines a and b. Line m also satisfies which of the following conditions?

(a) Line m is parallel to line a and b.

(b) Line m is skew to line a and b.

(c) Line m is perpendicular to plane R.

(d) Line m is parallel to plane R

Draw a picture or explain your answer in words.

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156 3

E

4

D 2P Q

X

A

B C

5.

Given: ; ;AD AE PX QX PD EQ

Prove: BD CE

15

A B

CD

M N

Mixed Review 2:

1. MN MN is the median of 2. Write the statement that is

Trapezoid ABCD. the inverse of ”If a quadrilateral is

a rhombus, then its’ diagonals are

perpendicular”? _________________________________

If AB=10, DC=14, then MN=_______

_________________________________

_________________________________

__________________________

__________________________

3. Two parallel lines below are cut by a transversal, find the value of x.

4. The measures of the angles of a quadrilateral are in the ratio 2:4:5:7.

Find the measure of the angles.

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5.

Given: DAC BCA

Prove: ABCD is a parallelogram

D C

BA

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Mixed Review 3:

1. 2.

3. Find the slope of a line that passes through the points (-6, 8) and (2, -4).

4.

5.

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Mixed Review 4:

1. 2.

3.

4. If the endpoints of the diameter of a circle are A (5, 2) and B (-3, 4), find the

coordinates of the center of the circle.

5. 6.

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Mixed Review 5:

1. If B, C and D are collinear, and m ACD = 50, what can you say about the

measure of angle ACB?

(1) 50m ACB (2) 50m ACB (3) 50m ACB (4) 40m ACB

2. What is the total number of points of intersection of the graphs of the

equations 2 5y x and y x . Draw a sketch to prove your answer.

3. Write the equation of the perpendicular bisector of line segment with

endpoints A(2, 6) and B(-2, 0).

4. Write the contrapositive of the statement:

“If you do your homework, then you will do well on the test”

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EA C

B5. Given:

ABC is an isosceles triangle with base AC

Segment BE is not a median

Prove:

Segment BE is not an angle bisector

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Parallelogram

- opp. sides || and

- diag. bisect each other

- opp s

Rectangle

- opp. sides || and

- diag. bisect each

other & are

- all right s

Square

- opp sides ||

- all sides

- diag. & bisect each other

- all right s

Rhombus

- opp. sides || and

- diag. bisect each other

- diag. bisect the s of rhom.

- all sides

Trapezoid

- 1 pr. opp sides ||

Thms. to prove

s are

ASA

SAS

AAS

SSS

HL (rt )

Thms. to prove s similar

AA similarity

SAS similarity

SSS similarity

Parallel lines

|| lines corresponding s

|| lines alternate interior s

|| lines alternate exterior s

|| lines same-side interior s supplementary

Coordinate Geometry

Distance = 2 2

2 1 2 1x x y y

Midpoint: 1 2 1 2,2 2

x x y y

slope = 2 1

2 1

y y

x x

slope-intercept form of line: y = mx + b

point-slope form of a line: 1 1y y m x x

circle equation: 2 2 2x h y k r

center: (h, k) and radius = r

parabola / quadratic equation:

2

2

; :2

; ,

by ax bx c axis of symmetry x

a

y a x h k vertex h k

Right Triangle

Trigonometry

sin

cos

tan

opp

hyp

adj

hyp

opp

adj

Altitude to Hypotenuse of Rt.

1

2

1

.

.

.

.

seg onhyp alt

alt seg on hyp

seg on hyp adj leg

adj leg whole hyp

Ratios for special

Right s

30-60-90 : 1 : 3 : 2

45-45-90 : 1 : 1 : 2

Logic

conditional: p q

converse: q p

inverse: ~ ~p q

contrapositive: ~ ~q p

Indirect Proof

Assume the opposite of “prove”

statement and continue a direct

proof method until there is a

contradiction (usually of the

given information)

Triangle Inequalities

- any 2 sides of have a sum greater than the

3rd

side

- the larger angle of a is opposite the larger

side of the

- exterior of a is greater than either remote

interior of the

22

l

cylinder

cone sphere

2

2

2 2

( )( )

SA r rh

V BaseArea height

V r h

2

13

213

( )( )

SA r rl

V BaseArea height

V r h

2

343

4SA r

V r

Circle Rules -- Angles:

1) central angle = meas. arc

2) inscribed angle = 1

2arc

3) formed by 2 chords = 1

2(sum arcs)

4) formed with vertex outside circle = 1

2 (difference arcs)

Circle Rules – Segments:

1) radius bisect chord chord to radius

2) 2 intersecting chords: products of the segments on

each chord are =

3) 2 secants:

(ext. seg)(whole secant) = (ext seg)(whole secant)

4) tangent/secant:

(tangent seg)2 = (ext. secant)(whole secant)

Triangles – by sides:

Scalene – all sides different lengths

Isosceles – 2 congruent sides

Equilateral – all 3 sides congruent

Triangles – by angles:

Acute – all angles are acute

Right – one right angle

Obtuse – one obtuse angle

Polygon Names:

Triangle – 3 sides

Quadrilateral – 4 sides

Pentagon – 5 sides

Hexagon – 6 sides

Octagon – 8 sides

Decagon – 10 sides

Regular Polygon: all sides

congruent and all interior

angles congruent

Polygon Angles – (n-sided)

Sum of interior s = 180 2n

Sum of exterior s = 360

Regular Polygons…

1 interior = 180( 2)n

n

1 exterior = 360

n SurfaceArea of Prism=(Perimeter)(height)+2(BaseArea)

SurfaceAreaofPyramid=(Perimeter)(slantheight)+BaseArea

Volume of Prism = (Base Area)(height)

Volume of Pyramid = 1

3(Base Area)(height)