Section 1: Introduction to Geometry – Points, Lines and Planes2

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Basics of Geometry – Part 1 What is geometry?

Geometry means “___________ __________________,” and it

involves the properties of points, lines, planes and figures.

What concepts do you think belong in this branch of mathematics?

Why does geometry matter? When is geometry used in the real world?

Points, lines, and planes are the building blocks of geometry.

Draw a representation for each of the following definitions and fill in the appropriate notation on the chart below.

Definition Representation Notation

A point is a precise location or place on a

plane. It is usually represented by a dot.

A line is a straight path that continues in both

directions forever. Lines are one-dimensional.

A line segment is a portion of a line

located between two points.

A ray is a piece of a line that starts at one point and extends infinitely in

one direction.

A plane is a flat, two-dimensional object. It has no

thickness and extends forever.

Unit 1

Name ________________________________

Period ____ Date ____

Section 1: Introduction to Geometry – Points, Lines and Planes3

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What can you say about multiple points on a line segment?

Section 1: Introduction to Geometry – Points, Lines and Planes4

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Try It!

2. Consider the definitions of points, lines, and planes youhave learned so far.

Use the word bank to complete the definitions below.Draw a representation of each one.

Word Bank

Parallel Planes ♦ Coplanar ♦ Parallel Lines Collinear ♦ Nonlinear

Definition

Points that lie on the same plane are

__________________.

Points that lie on the same line are

___________________.

Drawing

Section 1: Introduction to Geometry – Points, Lines and Planes5

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Try It!

2. Plane 𝒬𝒬 contains 𝐴𝐴𝐴𝐴 and 𝐵𝐵𝐵𝐵, and it also intersects 𝑃𝑃𝑃𝑃 onlyat point 𝑀𝑀.

Sketch plane 𝒬𝒬.

For points, lines, and planes, you need to know certain postulates.

Let’s examine the following postulates A through F.

A. Through any two points there is exactly one line.

B. Through any three non-collinear points there is exactly one plane.

C. If two points lie in a plane, then the line containing those points will also lie in the plane.

D. If two lines intersect, they intersect in exactly one point.

E. If two planes intersect, they intersect in exactly one line.

F. Given a point on a plane, there is one and only one line perpendicular to the plane through that point.

A postulate is a statement that we take to be automatically true. We do not need to prove that a postulate is true because it is something we assume to be true.

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, what is the value of

Section 1: Introduction to Geometry – Points, Lines and Planes8

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Try It!

3. Diego lives in a city and Anya lives in another city. Theirhouses are 72 miles apart. They both meet at their favoriterestaurant, which is 16𝑥𝑥 − 3 miles from Diego’s house and 5𝑥𝑥 + 2 miles from Anya’s house.

Diego argues that in a straight line distance, the restaurant is halfway between his house and Anya’s house. Is Diego right? Justify your reasoning.

Midpoint and distance can also be calculated on a coordinate plane.

The coordinate plane is a plane that is divided into ________ regions (called quadrants) by a horizontal line (______________) and a vertical line (______________).

Ø The location, or coordinates, of a point are given by an ordered pair, __________.

Section 1: Introduction to Geometry – Points, Lines and Planes9

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Section 1: Introduction to Geometry – Points, Lines and Planes10

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Try It!

6. 𝑃𝑃 has coordinates 2,4 . 𝑄𝑄 has coordinates (−10,12). Findthe midpoint of 𝑃𝑃𝑃𝑃.

7. Café 103 is equidistant and collinear from Metrics Schooland Angles Lab. The Metrics School is located at point4,6 on a coordinate plane, and Café 103 is at point 7,2 . Find the coordinates of Angles Lab.

Name ____________________ Unit 1 Supplement

Period _________

Page 1 of 4

The Pythagorean Theorem

Vocabulary: Pythagorean Theorem

In a right triangle, the square of the length of thehypotenuse is equal to the sum of the squares of thelengths of the legs.

Pythagorean triple

A set of three positive integers a, b, and c that satisfy the equation c2 = a2 + b2

Common Pythagorean Triples and Some of Their Multiples:

Converse of the Pythagorean Theorem

If the square of the length of the longest side of a triangle is equal to the sum of thesquares of the lengths of the other two sides, then the triangle is a right triangle.

If c2 = a2 + b2, then ΔABC is a right triangle.

Pythagorean Inequalities Theorem

For any ΔABC, where c is the length of the longest side of the triangle, the followingstatements are true:

If c2 < a2 + b2, then ΔABC is acute If c2 > a2 + b2, then ΔABC is obtuse

10a

Name ____________________

Period _________

Page 2 of 4

Example 1: Using the Pythagorean Theorem

Find the value of x. Decide whether the side lengths form a Pythagorean triple.

Pythagorean Triple Yes/No

x = ______

Example 2: Using the Pythagorean Theorem

Find the value of x. Decide whether the side lengths form a Pythagorean triple.

Pythagorean Triple Yes/No

x = ______

Example 3: Solving a Real-Life Problem

Jennie rides her bike 4 kilometers east and then 2 kilometers north. Use the Pythagorean Theorem to find how far she is from her starting point. Round your answer to the nearest hundredth of a kilometer.

Distance = __________

Unit 1 Supplement

The Pythagorean Theorem

10b

Name ____________________

Period _________

Page 3 of 4

Example 4: Solving a Real-Life Problem

An anemometer is a device used to measure wind speed. The anemometer shown is attached to the top of a pole. Support wires are attached to the pole 5 feet above the ground. Each support wire is 6 feet long. How far from the base of the pole is each wire attached to the ground?

Distance = __________

Example 5: Converse of the Pythagorean Theorem

a) Determine whether the triangle is a right triangle

Right Triangles? Yes/No

b) Determine whether the triangle is a right triangle

Right Triangles? Yes/No

Unit 1 Supplement

The Pythagorean Theorem

10c

Name ____________________

Period _________

Page 4 of 4

Example 6: Classifying Triangles

Verify that segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form a triangle. Is the triangle acute, right, or obtuse?

Forms a Triangle? Yes/No Acute/Right/Obtuse?

a2 = _______

b2 = _______

c2 = _______

Example 7: Classifying Triangles

Verify that segments with lengths of 3 feet, 4 feet, and 6 feet form a triangle. Is the triangle acute, right, or obtuse?

Forms a Triangle? Yes/No Acute/Right/Obtuse?

a2 = _______

b2 = _______

c2 = _______

Example 8: Classifying Triangles

Verify that segments with lengths of 2.1 feet, 2.8 feet, and 3.5 feet form a triangle. Is the triangle acute, right, or obtuse?

Forms a Triangle? Yes/No Acute/Right/Obtuse?

a2 = _______

b2 = _______

c2 = _______

Unit 1 Supplement

The Pythagorean Theorem

10d

Section 1: Introduction to Geometry – Points, Lines and Planes11

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Section 1: Introduction to Geometry – Points, Lines and Planes14

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The following formula can be used to find the coordinates of a given point that partition a line segment into ratio 𝑘𝑘.

𝑥𝑥,𝑦𝑦 = 𝑥𝑥! + 𝑘𝑘 𝑥𝑥! − 𝑥𝑥! ,𝑦𝑦! + 𝑘𝑘 𝑦𝑦! − 𝑦𝑦!

Let’s Practice!

1. What is the value of 𝑘𝑘 used to find the coordinates of apoint that partitions a segment into a ratio of 4:3?

2. Determine the value of 𝑘𝑘 if partitioning a segment into aratio of 1:5.

Section 1: Introduction to Geometry – Points, Lines and Planes15

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Section 1 – Topic 6 Partitioning a Line Segment – Part 2

Consider M, N, and P, collinear points in MP.

What is the difference between the ratio 𝑀𝑀𝑀𝑀:𝑁𝑁𝑁𝑁 and the ratio of 𝑀𝑀𝑀𝑀:𝑀𝑀𝑀𝑀?

What should you do if one of the parts of a ratio is actually the whole line instead of a ratio of two smaller parts or segments?

Let’s Practice!

1. Points 𝑃𝑃,𝑄𝑄, and 𝑅𝑅 are collinear on 𝑃𝑃𝑃𝑃, and 𝑃𝑃𝑃𝑃:𝑃𝑃𝑃𝑃 = !!. 𝑃𝑃 is

located at the origin, 𝑄𝑄 is located at (𝑥𝑥,𝑦𝑦), and 𝑅𝑅 is located at (−12,0). What are the values of 𝑥𝑥 and 𝑦𝑦?

Section 1: Introduction to Geometry – Points, Lines and Planes16

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Section 1: Introduction to Geometry – Points, Lines and Planes17

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m =

Section 1: Introduction to Geometry – Points, Lines and Planes18

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Try It!

2. Match each of the following with the equations below.Write the letter of the appropriate equation in the columnbeside each item.

A. 𝑥𝑥 = −5 B. 𝑦𝑦 = − !!𝑥𝑥 + 1 C. 3𝑥𝑥 − 5𝑦𝑦 = −30 D. 𝑥𝑥 − 2𝑦𝑦 = −2

A line parallel to 𝑦𝑦 = !!𝑥𝑥 + 2

A line perpendicular to 𝑦𝑦 = 4

A line perpendicular to 4𝑥𝑥 + 2𝑦𝑦 = 12

A line parallel to 2𝑥𝑥 + 8𝑦𝑦 = 7

Let’s Practice!

1. Indicate whether the lines are parallel, perpendicular, orneither. Justify your answer.

a. 𝑦𝑦 = 2𝑥𝑥 and 6𝑥𝑥 = 3𝑦𝑦 + 5

b. 2𝑥𝑥 − 5𝑦𝑦 = 10 and 10𝑥𝑥 + 4𝑦𝑦 = 20

c. 4𝑥𝑥 + 3𝑦𝑦 = 63 and 12𝑥𝑥 − 9𝑦𝑦 = 27

d. 𝑥𝑥 = 4 and 𝑦𝑦 = −2

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Section 1 – Topic 8 Parallel and Perpendicular Lines – Part 2

Let’s Practice!

1. Write the equation of the line passing through (−1, 4) andperpendicular to 𝑥𝑥 + 2𝑦𝑦 = 11.

Try It!

2. Suppose the equation for line 𝐴𝐴 is given by 𝑦𝑦 = − !!𝑥𝑥 − 2. If

line 𝐴𝐴 and line 𝐵𝐵 are perpendicular and the point (−4, 1)lies on line 𝐵𝐵, then write an equation for line B.

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2. A parallelogram is a four-sided figure whose opposite sidesare parallel and equal in length. Alex is drawingparallelogram 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 on a coordinate plane. Theparallelogram has the coordinates 𝐴𝐴(4,2), 𝐵𝐵(0,−2), and𝐷𝐷(8,−1).

Which of the following coordinates should Alex use for point 𝐶𝐶 ?

A 6,−3B 4,−5C 10,−3D 4,3

BEAT THE TEST!

1. The equation for line 𝐴𝐴 is given by 𝑦𝑦 = − !!𝑥𝑥 − 2. Suppose

line 𝐴𝐴 is parallel to line 𝐵𝐵, and line 𝑇𝑇 is perpendicular to line

𝐴𝐴. Point (0,5) lies on both line 𝐵𝐵 and line 𝑇𝑇.

Part A: Write an equation for line 𝐵𝐵.

Part B: Write an equation for line 𝑇𝑇.

End S1