Lesson 15.0

UNIT 15

INTRODUCTION TO GEOMETRY

Review. . . . . . .

Lesson 15.1 Points, Lines, and Planes

Lesson 15.2 Congruent Segments

Lesson 15.3 Angles . . . .

Lesson 15.4 Parallel and Skew Lines

Lesson 15.5 Triangles

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Lesson 15.0 Review

Radical Equations:

Unit 15

A radical equation is one which contains one or more radicals involving the variable in the radicand. To solve a radical equation:

'12x + 1 - 3 = 2

'12x + 1 = 5

2x + 1 = 25

2x = 24

X - 12

Solving x2 = a or (kx + b)2 = a:

5(4x - 1)2 = 20

(4x - 1)2 = 4

• Get a single rad.teal tennfor one side.

• Square both sides. (Repeat steps 1 and 2 lf a rad.teal still appears.)

• Solve the resulting (radical-free) equation.

• Answer. Check solutionf s) tn original equation.

• Simplify to get a perfect square member.

4x - 1 = 2 or 4x - 1 = -2 • Extract roots: then solve each equation.

4x = 3 or

x = 3 or 4

4x= -1

X = _ j_ 4

• Answer. Check both solutions in original

equation.

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Completing the Square:

x2 - 6x + 5 = 0

x2 - 6x = -5

x2 - 6x + 9 = 4

(x - 3)2 = 4

x - 3 = 2 or x - 3 = -2

x = 5 or X = 1

Quadratic Formula:

Lesson 15.0

• Left side ts not a perfect square.

• Keep variable tem1S on one side, constant on the other.

• Add 9 (square of half the linear term

coeffic(en.tJ=( {) 2 = (-3) 2 = 9.

• Write trinomtal as square of a binomial

• Extract roots: then solve each equation.

• Answer. Check both roots in original equation.

To solve equations of the form ax2 +bx+ c = 0, use the quadratic formula:

2x2 = 3x - 1

2x2 - 3x + 1 = 0

x = -b ± ...Jb2 - 4ac 2a

x = - (-3) ±. y(-3)2 - 4{2Hll 2(2)

x=3±y9-8 4

X ~ 3 ±-fl 4

x = 3 ± 1 = 1 or 1. 4 2

• Standardform: a= 2, b = -3, c = 1.

• Substi.tute values for "a", "b," and "c" tnto quadratic formula.

• Simplify.

• Check both answers.

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Lesson 15.0

The Parabola:

y = ax2 + bx + c, a '# 0 has a parabola for its graph.

y = x2 + 2x - 3

0 = x2 + 2x - 3

X = -2 ± ..J22 - 4(1)(-3) 2(1)

X = -2 ± fT6 = -2 ± 4 2 2

x = 1 or x = -3

X = 1 + (-3) 2

X = -1

y = (-1)2 + 2(-1) - 3

y=l-2-3

y = -4

Graph of y = x2 + 2x - 3:

• "a" is + (+ 1), so graph opens upward. If "a" is -. graph wlll open. downward.

• Let y = O: solve for "x' wf1Ild x-intercepts.

• x-tntercepts of lhe parabola.

• Find axis of symmetry.

• Axis of symmetry is x = -1 (vertical line half -way between x-fntercepts).

• Substitute -1 for "x' to fmd vertex.

• Vertex is at (-1, -4).

y

axis of symmetr

X=-1

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x-intercepts: -3 and 1

y-intercept: (0,-3) vertex: (-1,-4)

Review Problems

Solve each equation:

1. '12x - 6 = 4 + -./x

2. (2x + 4)2 = 81

Solve by completing the square:

5. x2 + 4x = -3

3. Y3x + 1 = Y5x - 9

4. 2(3x - 5)2 + 6 = 14

6. x2 - 1 Ox + 9 = 0

Solve by using the quadratic formula:

7. x2 - 7x + 12 = 0 8. 2x2 - 16x + 30 = 0

Write an equation and solve:

Lesson 15.0

9. Toe base of a triangle is three times the height. If the area of the triangle is 37.5m 2• find the base and height of the triangle.

Graph the equation. Find the x-intercepts, axis of symmetry, and vertex. Tell if the vertex is a maximum or mfnfm,un:

10. y = x2 + 6x - 5

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Lesson 15.1 Points, Lines, and Planes Unit ·15

Rules:

Example:

Rule:

The study of points is called geometry. The study of geometry must begin with three words that will remain undefined. These undefined terms are point, line, and plane. These three terms serve as the base for defming other geometric terms.

Shown in the chart below is the information needed to identify point, line and plane:

Undefined Pictoral Other Term Representation Label Facts

A point has no

POINT . • p length, width, or thickness.

A dot. Capital letter

•x v• A line has length • • .... .... 1out no width or

LINE Straight line XY YX lthickness. It is

with arrows Two capital infinite in length.

on each end. letters or a It is defined by '"'·'-...... ,...~~~ I? ~,~t,nrt .-.~ln+~

7 L 7 IA plane has length

L m 10ut no thickness . . PLANE Parallelogram Lower-case It is flat and in-

letter. inite. It is defined ov 3 ooints.

1. Give seven possible names for the line: ~ t t t t .b

C D E ~~._.-.-c-+._. CD, CE, DE, DC, EC, ED, orb.

C

In geometry, a good definition uses only previously accepted undefined terms or previously defined terms. Definitions:

1. Space is the set of all points. 2. A geometric figure is a set of points. 3. Points that lie on the same line are said to be collinear

points. Noncollinear points do not lie on the same line. 4. Points that lie in the same plane are said to be coplanar.

Noncoplanar points do not lie in the same plane.

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Lesson 15.1

Example: 2. Find four collinear points, three coplanar points, three noncollinear points, and four noncoplanar points:

F, E, D, and H are collinear.

D, E, and Hare coplanar.

G, E, and H are noncollinear.

F, H, and Gare noncoplanar.

HO:MEWORK

Identify the following points as collinear, coplanar, noncollinear, or noncoplanar:

1. BCE

2. ABC

3. DFE

4. ABE

5. CFB

Draw each figure as it is described. Label the points, lines and planes: ~

6. AB intersecting planed ~

7. CD lying on plane r <-7>

8. GH lying on plane l with point Ron the plane but noncollinear with <-> GH

9. tit intersecting plane eat point B, and DB coplanar to plane e

10. Point C on plane f with DE intersecting plane f

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Lesson 15.2 Congruent Segments Unit 15

Rule: If two segments are the same length, they are congruent segments. = is the symbol for congruent.

Example: 1. A B C D

Rule:

----- -----2 3 4 5 6

• AB~ CD because they both have the same length.

The midpoint of a segment is the point that bisects the segment.

Example: 2. _________ _ • Point M is the midpoint which bisects AB if AM~ MB.

Rule:

A M B

To construct a segment that is congruent to another segment, a compass must be adjusted to the distance from the two end points of the original segment. Then the compass is moved to the new segment that is then marked by the compass.

Example: 3. Construct a segment congruent to a given segment:

Given: LM Construct: NO = LM on line r

L M

Solution: NO = LM

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• Adjust the compass to the distance of IM.

• Move the compass to line "r," keeping the same opening.

• Use the compass to mark points N and O on line "r."

----- -- ----- - -- - --- -

Lesson 15.2

HOMEWORK

Which pairs of segments are congruent? Use a ruler to measure each pair of segments:

1. 2. 3. 4. 5.

Which pairs of segments are congruent? Use a compass to compare the lengths:

7. 8. 9.

Which pairs of segments are conanient? Use the coordinates of the number line to decide:

A B C D E F G H I J K L MN 0 p Q R ST UV W X Y

• I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

11. AC and IK 14. GR and WL 12. BG and MS 15. AX and YC 13. XM and AL

I I 14

Construct a segment congruent to the given segment. Next, construct a mi~point on each of the new segments:

16. A B

19. G H

17. C D 20.

18. E

J F

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•

Lesson 15.3 Angles Unit 15

Rule: The union of two rays with a common end point is called an angle. The symbol L is used for angle.

Example: 1. Draw an angle:

Rules:

Acute Angle

• Air and 'Xe have a common end point A.

A C Every angle has a vertex and two sides:

A There are four basic types of angles:

1. Acute angles have a degree measure that is greater than o0 and less than goo.

2. Right angles have a goo measure. 3. Obtuse angles have a degree measure that is greater

than goo and less than 180°. 4. Straight angles have a degree measure of 180°.

• Right Angle Obtuse Angle Straight Angle

Example: 2. Label each angle as an acute, right, obtuse or straight angle:

A. 8. C. D.

Obtuse Angle

Acute Angle

Right Angle Obtuse Angle

E.

•

Straight Angle •

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Lesson 15.3

Rule: Angles can be labeled in several different ways, using points on the rays:

Example: 3. L BAC LCAB LA ~x

Draw the given angle:

1. acute angle

2. straight angle

3. right angle

4. obtuse angle

A C

HOMEWORK

Label all the angles in each drawing. Use all four methods:

8 . 5. C • I

A L

B C

9.

6.

d

b R

D E

10. 7.

s

b

y z L D

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I • R

s

Lesson 15.4 Parallel and Skew Lines Unit 15

Rules: Some lines intersect while other lines do not.

Lines that do not intersect may either be in the same plane or in different planes.

Lines that lie in the same plane (coplanar) that do not intersect are parallel lines.

Lines that do not lie in the same plane (noncoplanar) are skew lines.

Examples: 1.

Rule:

s u

T V

2. A

B

D

C

3.

._,..~ ++ • ST II UV means ST is parallel to1.N.

~ ~-> ~ • AC-Ir BD Tr]£_ans AC is not

parallel to 13-a

• 11tese lines are skew because they do not lie in the same plane.

Segments of lines and rays may also be parallel if the lines which contain them are parallel.

Example: 4. Which segments are parallel?

A. B.

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4.

1 o.

Lesson 15.4

Example: 5. Which rays are parallel?

E.~

F .

• HOMEWORK

Which pairs of lines, rays or segments are parallel? Which are skew? Which are neither parallel nor skew?

3.

5. 6.

8.

t

1/ 11 .

• 4

12.

13. Define parallel lines.

14. Define skew lines.

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Lesson 15.5 Triangles Unit 15

Rules:

Rules:

In geometry, one of the basic figures that is studied is a triangle. A triangle is a figure with three segments that have end points in common. The three end points must be non collinear.

The sides of the triangle are formed by the line segments. The vertices are the three end points.

vertex

vertex vertex side

Triangles are either classified by the length of their sides or the measure of their angles.

Classifications by Length qf Sides

1. Equilateral triangle: All three sides are congruent.

2. Isosceles triangle:

3. Scalene triangle:

At least two sides are congruent.

There are no congruent sides.

Example: 1. Draw an equilateral triangle, an isosceles triangle, and a scalene triangle:

A. B. C.

Equilateral Triangle Isosceles Triangle Scalene Triangle

Classifications by Measure of Angle

Rules: 1. · Acute triangle: Three acute angles.

2. Right triangle: One right angle.

3. Obtuse triangle: One obtuse angle.

4. Equiangular triangle: Three congruent angles.

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------ ------~-- - - - ·-- · ---

Lesson 15.5

Example: 2. Draw an acute triangle, right triangle, obtuse triangle, and equiangular triangle:

D.D. E. ~

Acute Triangle Right Triangle

HOMEWORK

Classify each trlangle by its sides:

Classify each tr1angle by lta angle•:

6.

Classify each trlangle by its sides and angles:

11. 12. 13.

125

F.~ G.D Obtuse Triangle Equiangular Triangle

14. 15.

Table of Roots and Powers

Sq. Cu. Sq . Cu. No. so, Root Cube Root No. Sq . Root Cube Root

1 1 1,000 1,000 51 2,601 7 I 141 132,651 3,708 2 4 1, 414 8 1,260 52 2,704 7, 211 140,608 3,733 3 9 1, 732 27 1,442 53 2,809 7,280 148,877 3,756 4 , 16 2,000 64 1,587 54 2,916 7,348 157,564 3,780 5 25 2,236 125 1, 71 0 55 3,025 7,416 166,375 3,803

. 6 36 2,449 216 1,817 56 3, 136 7,483 175,616 3,826 7 49 2,646 343 1 , 913 57 3,249 7,550 185,193 3,849 8 64 2,828 512 2,000 58 3,364 7,616 195,112 3,871 9 81 3,000 729 2,080 59 3,481 7,681 205,379 3,893

10 100 3,162 1,000 2, 154 60 3,600 7,746 216,000 3,915 11 121 3,317 1 , 331 2,224 61 3,721 7,810 226,981 3,936 12 144 3,464 1,728 2,289 62 3,844 7,874 238,328 3,958 13 169 3,606 2,197 2,351 63 3,969 4,937 250,047 3,979 14 196 3,742 2,744 2,410 64 4,096 8,000 262,144 4,000 15 225 3,873 3,375 2,466 65 4,225 8,062 274,625 4,021 16 256 4,000 4,096 2,520 66 4,356 8 I 124 287,496 4,041 17 289 4,123 4,913 2,571 67 4,489 8 I 185 300,763 4,062 18 324 4,243 5,832 2,621 68 4,624 8,246 314,432 4,082 19 361 4,359 6,859 2,668 69 4,761 8,307 328,509 4,102 20 400 4,472 8,000 2,714 70 4,900 8,357 343,000 4, 121 21 441 4,583 9,261 2,759 71 5,041 8,426 357,911 4, 141 22 484 4,690 10,648 2,802 72 5,184 8,485 373,246 4,160 23 529 4,796 12,167 2,844 73 5,329 8,544 389,017 4,179 24 576 4,899 13,824 2,884 74 5,476 6,602 405,224 4,198 25 625 5,000 15,625 2,924 75 5,625 6,660 421,875 4,217 26 676 5,099 17,576 2,962 76 5,776 8,718 438,976 4,236 27 729 5,196 19,663 3,000 77 5,929 8,775 456,533 4,254 28 784 5,292 21,952 3,037 78 6,084 6,832 474,552 4,273 29 841 5,365 24,389 3,072 79 6,241 8,888 . 493,039 4,291 30 900 5,477 27,000 3,107 80 6,400 8,944 512,000 4,309 31 961 5,566 29,791 3,141 81 6,561 9,000 531,441 4,327 32 1 , 0 24 5,657 32,768 3, 175 82 6,724 9,055 551,368 4,344 33 1,089 5,745 35,937 3,208 83 6,889 9 I 11 Q 571,787 4,362 34 1 , 156 5,831 39,304 3,240 84 7,056 9 I 165 592,704 4,380 35 1,125 5,916 42,875 3,271 85 7,225 9,220 614,125 4,397 36 1,296 6,000 46,656 3,302 86 7,396 9,274 636,056 4,414 37 1,369 6,083 50,653 3,332 87 7,569 9,327 658,503 4,431 38 1,444 6, 164 54,872 3,362 88 7,744 9,381 681,472 4,448 39 1 , 5 21 6,245 59,319 3,391 89 7,921 9,434 704,969 4,465 40 · 1,600 6,325 64,000 3,420 90 8,100 9,487 729,000 4,481 41 1,681 6,403 68,921 3,448 91 8,281 9,539 753,571 4,498

42 1, 764 6,481 74,088 3,476 92 8,464 9 I 592 778,688 4,514

43 1,849 6,557 79,507 3,503 93 8,649 9,644 804,357 4,531 44 1,936 6,633 85,184 3,530 94 8,836 9,695 830,584 4,547 45 2,025 6,708 91,125 3,557 95 9,025 9,747 857,375 4,563

46 2, 11 6 6,782 97,336 3,583 96 9,216 9,798 884, 736 4,579 47 2,209 6,856 103,823 3,609 97 9,409 9,849 912,673 4,595 48 2,304 6,928 110,592 3,634 98 9,604 9,899 941,192 4,610 49 2,401 7,000 117,649 3,659 99 9,801 9,950 970,299 4,626

so 2,500 7,071 125,000 3,684 100 10,000 10,000 1,000,000 4,642