Assume there is a line (l) passing thru two distinct points (P) and (Q). In this case we might denote (l) as ((PQ)). There might be more than one line thru (P) and (Q), but if we write ((PQ)) we assume that we made a choice of such line.

We will denote by ([PQ)) the *half-line* that starts at (P) and contains (Q). Formally speaking, ([PQ)) is a subset of ((PQ)) which corresponds to ([0,infty)) under an isometry (f: (PQ) o mathbb{R}) such that (f(P) = 0) and (f(Q) > 0).

The subset of line ((PQ)) between (P) and (Q) is called the *segment* between (P) and (Q) and denoted by ([PQ]). Formally, the segment can be defined as the intersection of two half-lines: ([PQ] = [PQ) cap [QP)).

Exercise (PageIndex{1})

Show that

(a) if (X in [PQ)), then (QX = |PX - PQ|);

(b) if (X in [PQ]), then (QX + XQ = PQ).

- Hint
Fix an isometry (f: (PQ) o mathbb{R}) such that (f(P) = 0) and (f(Q) = q > 0).

Assume that (f(X) = x). By the definition of the half-line (X in [PQ)) if and only if (x ge 0). Show that the latter holds if and only if (|x - q| = ||x| - |q||). Hence (a) follows.

To prove (b), observe that (X in [PQ]) if and only if (0 le x le q). Show that the latter holds if and only if (|x - q| + |x| = |q|).

## Lesson 6

Which of the following constructions would help to construct a line passing through point (C) that is perpendicular to the line (AB) ?

Expand Image

Construction of an equilateral triangle with one side (AB)

Construction of a hexagon with one side (BC)

Construction of a perpendicular bisector through (C)

Construction of a square with one side (AB)

### Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

### Problem 2

Two distinct lines, (ell) and (m) , are each perpendicular to the same line (n) . Select **all** the true statements.

Lines (ell) and (m) are perpendicular.

Lines (ell) and (n) are perpendicular.

Lines (m) and (n) are perpendicular.

Lines (ell) and (m) are parallel.

Lines (ell) and (n) are parallel.

Lines (m) and (n) are parallel.

### Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

### Problem 3

This diagram is a straightedge and compass construction of the bisector of angle (BAC) . Only angle (BAC) is given. Explain the steps of the construction in order. Include a step for each new circle, line, and point.

Expand Image

**Description:** <p>Three circles intersect. Large circle center A. Two congruent small circles with center B and D. Large circle intersects points B and D. Circle center B intersects points E, D and F. Circle center D intersects points E, B and F. Segment A C passes through D, Segment A F passes through E, Segment A B is drawn.</p>

### Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

### Problem 4

This diagram is a straightedge and compass construction of a line perpendicular to line (AB) passing through point (C) . Which segment has the same length as segment (EA) ?

Expand Image

**Description:** <p>Three circles. Two large congruent circles intersect at point E. Smaller circle, center C, in the intersection of two larger circles. Small circle intersects one circle at Point A and the other circle at Point D. Segment A B passes through C and D. Segment E C is drawn.</p>

### Solution

### Problem 5

This diagram is a straightedge and compass construction. Which triangle is equilateral? Explain how you know.

Expand Image

**Description:** <p>Seven congruent circles, one centered circle with six circles on it. Center points labeled Z, W, and T. Two circles intersect at top point V and two circles intersect at bottom point U. One large equilateral triangle Z V W and one small isosceles triangle S T U formed.</p>

### Solution

### Problem 6

In the construction, (A) is the center of one circle, and (B) is the center of the other. Name the segments in the diagram that have the same length as segment (AB) .

Expand Image

### Solution

### Problem 7

This diagram is a straightedge and compass construction. (A) is the center of one circle, and (B) is the center of the other.

- Name a pair of perpendicular line segments.
- Name a pair of line segments with the same length.

Expand Image

**Description:** <p><strong>Two circles intersect. Large circle center A. Smaller circle center C, goes through center A and intersects larger circle at point B. Point D on smaller circle. Segment A D passes through C. Segments A B, C B, and D B are drawn.</strong></p>

### Solution

### Problem 8

(A) , (B) , and (C) are the centers of the 3 circles. Select **all** the segments that are congruent to (AB) .

Expand Image

**Description:** <p>Three congruent, intersecting circles, each pass through the others center at points A, B and C. Circles center A and C intersect at points H and B. Circles center A and B intersect at points D and C. Circles center B and C intersect at points F and A. Segment D H passes through A. Segment D F passes through B, Segment F H passes through C, Segments C D and A B intersect at point E.<br> </p>

## 1.6: Half-lines and segments - Mathematics

**3847** days since **Winter Break**

#### TEACHERS-STUDENTS RESOURCES

### Lesson 2: 1-2 Points, Lines and Planes

**OVERVIEW/ Expectations**

In this lesson, students will gain a basic understanding of geometric terms and postulates.

The students will analyze two-and three- dimensional figures using tools and technology when appropriate.

Cadets will be able to Identify points, lines, rays, and segments.

Cadets will be able to identify, name, and draw points, lines, segments,rays , and planes.

Solve problems involving points, lines, and planes

in order to solve real-world problems

** ENDURING UNDERSTANDINGS** :

Points , lines , and planes are the foundations of geometry

** ESSENTIAL QUESTIONS:**

Why are point, line, and plane the undefined terms of geometry?

How are properties of geometric figures related to their measurable attributes?

What are the undefined terms in geometry and how do we use them?

How do we sketch the intersections of lines and planes?

1.0. Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning

2.1.1 analyze properties of geometric figures.

2.1.1.a identify and describe the basic undefined terms of geometry

2.1.1.c. Represent and analyze point relationships including collinear and coplanar

2.1.4 construct and/or draw and/or validate properties of geometric figures using appropriate tools and technology

** INSTRUCTIONAL SEQUENCE**

1.1. The Building Blocks of Geometry

Definition that results from the undefined terms

Investigations of postulates that result from the undefined terms

Introduction to dynamic geometry software

1.4. Geometry using Paper Folding

1.5. Special Points in a triangles

1.7. Motion in the coordinate Plane

To which part of your curriculum does this lesson relate? State standard 1: Make sense of problems and persevere in solving them. State standard 2: Reason abstractly and quantitatively State standard 4: Model with mathematics State standard 5: Use appropriate tools strategically State standard 6:Attend to precision State standard 7:Look for and make use of structure State standard 8:Look for and express regularity in repeated reasoning

**Axiom** - A statement about real numbers that is accepted as true without proof.

**Postulate** - A statement about geometry that is accepted as true without proof.

**Theorem** - A statement in geometry that has been proved.

**Undefined Terms** - Terms that aren't defined, but instead explained they form the foundation for all definitions in geometry. The undefined terms point, line, and plane are the building blocks of geometry.

The term point, line, and plane are considered undefined because they cannot be defined in terms of other figures.

Segment, ray, and angle are defined in terms of the undefined terms and each other

Undefined term, point, line, plane, collinear, coplanar , segment , endpoint , ray, opposite ray , postulate .

## Lesson 6

Which of the following constructions would help to construct a line passing through point (C) that is perpendicular to the line (AB) ?

Expand Image

Construction of an equilateral triangle with one side (AB)

Construction of a hexagon with one side (BC)

Construction of a perpendicular bisector through (C)

Construction of a square with one side (AB)

### Solution

For access, consult one of our IM Certified Partners.

### Problem 2

Two distinct lines, (ell) and (m) , are each perpendicular to the same line (n) . Select **all** the true statements.

Lines (ell) and (m) are perpendicular.

Lines (ell) and (n) are perpendicular.

Lines (m) and (n) are perpendicular.

Lines (ell) and (m) are parallel.

Lines (ell) and (n) are parallel.

Lines (m) and (n) are parallel.

### Solution

For access, consult one of our IM Certified Partners.

### Problem 3

This diagram is a straightedge and compass construction of the bisector of angle (BAC) . Only angle (BAC) is given. Explain the steps of the construction in order. Include a step for each new circle, line, and point.

Expand Image

**Description:** <p>Three circles intersect. Large circle center A. Two congruent small circles with center B and D. Large circle intersects points B and D. Circle center B intersects points E, D and F. Circle center D intersects points E, B and F. Segment A C passes through D, Segment A F passes through E, Segment A B is drawn.</p>

### Solution

For access, consult one of our IM Certified Partners.

### Problem 4

This diagram is a straightedge and compass construction of a line perpendicular to line (AB) passing through point (C) . Which segment has the same length as segment (EA) ?

Expand Image

**Description:** <p>Three circles. Two large congruent circles intersect at point E. Smaller circle, center C, in the intersection of two larger circles. Small circle intersects one circle at Point A and the other circle at Point D. Segment A B passes through C and D. Segment E C is drawn.</p>

### Solution

For access, consult one of our IM Certified Partners.

### Problem 5

This diagram is a straightedge and compass construction. Which triangle is equilateral? Explain how you know.

Expand Image

**Description:** <p>Seven congruent circles, one centered circle with six circles on it. Center points labeled Z, W, and T. Two circles intersect at top point V and two circles intersect at bottom point U. One large equilateral triangle Z V W and one small isosceles triangle S T U formed.</p>

### Solution

For access, consult one of our IM Certified Partners.

### Problem 6

In the construction, (A) is the center of one circle, and (B) is the center of the other. Name the segments in the diagram that have the same length as segment (AB) .

Expand Image

### Solution

For access, consult one of our IM Certified Partners.

### Problem 7

This diagram is a straightedge and compass construction. (A) is the center of one circle, and (B) is the center of the other.

- Name a pair of perpendicular line segments.
- Name a pair of line segments with the same length.

Expand Image

**Description:** <p><strong>Two circles intersect. Large circle center A. Smaller circle center C, goes through center A and intersects larger circle at point B. Point D on smaller circle. Segment A D passes through C. Segments A B, C B, and D B are drawn.</strong></p>

### Solution

For access, consult one of our IM Certified Partners.

### Problem 8

(A) , (B) , and (C) are the centers of the 3 circles. Select **all** the segments that are congruent to (AB) .

Expand Image

**Description:** <p>Three congruent, intersecting circles, each pass through the others center at points A, B and C. Circles center A and C intersect at points H and B. Circles center A and B intersect at points D and C. Circles center B and C intersect at points F and A. Segment D H passes through A. Segment D F passes through B, Segment F H passes through C, Segments C D and A B intersect at point E.<br> </p>

### Solution

For access, consult one of our IM Certified Partners.

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

This book includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information.

## Lesson 6

In this lesson, students use previous constructions to create new constructions. This provides an example of a more general theme of learning—that discoveries build on one another and lay the foundation for new knowledge. The subsequent sections and units will guide students through a process of establishing a theoretical foundation and building new knowledge on itself within the framework of transformational geometry. Students make use of the structure that two circles of the same radius that go through each other’s center can be used to construct perpendicular lines to think strategically about how to construct a line perpendicular to a given line that goes through a given point *not* on the line (MP7). As students continue to apply the method for constructing a perpendicular line to construct a parallel line, they are engaging in repeated reasoning (MP8).

In the warm-up, students identify rigid motions as a review of their study of congruence in grade 8. In the cool-down, students use a construction to reflect a point across a line—although they may not realize that was what they did—as a preview of subsequent lessons.

## Contents

If *V* is a vector space over R *L* is a subset of *V*, then *L* is a **line segment** if *L* can be parameterized as

Sometimes, one needs to distinguish between "open" and "closed" line segments. In this case, one would define a **closed line segment** as above, and an **open line segment** as a subset *L* that can be parametrized as

Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.

In geometry, one might define point *B* to be between two other points *A* and *C*, if the distance *AB* added to the distance *BC* is equal to the distance *AC*. Thus in R 2 *A* = (*a _{x}*,

*a*) and

_{y}*C*= (

*c*,

_{x}*c*) is the following collection of points:

_{y}

- A line segment is a connected, non-emptyset.
- If
*V*is a topological vector space, then a closed line segment is a closed set in*V*. However, an open line segment is an open set in*V*if and only if*V*is one-dimensional. - More generally than above, the concept of a line segment can be defined in an ordered geometry.
- A pair of line segments can be any one of the following: intersecting, parallel, skew, or none of these. The last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane then they must cross each other, but that need not be true of segments.

In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or defined in terms of an isometry of a line (used as a coordinate system).

Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets, to the analysis of a line segment. The segment addition postulate can be used to add congruent segment or segments with equal lengths, and consequently substitute other segments into another statement to make segments congruent.

A line segment can be viewed as a degenerate case of an ellipse, in which the semiminor axis goes to zero, the foci go to the endpoints, and the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant if this constant equals the distance between the foci, the line segment is the result. A complete orbit of this ellipse traverses the line segment twice. As a degenerate orbit, this is a radial elliptic trajectory.

In addition to appearing as the edges and diagonals of polygons and polyhedra, line segments also appear in numerous other locations relative to other geometric shapes.

### Triangles Edit

Some very frequently considered segments in a triangle to include the three altitudes (each perpendicularly connecting a side or its extension to the opposite vertex), the three medians (each connecting a side's midpoint to the opposite vertex), the perpendicular bisectors of the sides (perpendicularly connecting the midpoint of a side to one of the other sides), and the internal angle bisectors (each connecting a vertex to the opposite side). In each case, there are various equalities relating these segment lengths to others (discussed in the articles on the various types of segment), as well as various inequalities.

Other segments of interest in a triangle include those connecting various triangle centers to each other, most notably the incenter, the circumcenter, the nine-point center, the centroid and the orthocenter.

### Quadrilaterals Edit

In addition to the sides and diagonals of a quadrilateral, some important segments are the two bimedians (connecting the midpoints of opposite sides) and the four maltitudes (each perpendicularly connecting one side to the midpoint of the opposite side).

### Circles and ellipses Edit

Any straight line segment connecting two points on a circle or ellipse is called a chord. Any chord in a circle which has no longer chord is called a diameter, and any segment connecting the circle's center (the midpoint of a diameter) to a point on the circle is called a radius.

In an ellipse, the longest chord, which is also the longest diameter, is called the *major axis*, and a segment from the midpoint of the major axis (the ellipse's center) to either endpoint of the major axis is called a *semi-major axis*. Similarly, the shortest diameter of an ellipse is called the *minor axis*, and the segment from its midpoint (the ellipse's center) to either of its endpoints is called a *semi-minor axis*. The chords of an ellipse which are perpendicular to the major axis and pass through one of its foci are called the latera recta of the ellipse. The *interfocal segment* connects the two foci.

When a line segment is given an orientation (direction) it is called a **directed line segment**. It suggests a translation or displacement (perhaps caused by a force). The magnitude and direction are indicative of a potential change. Extending a directed line segment semi-infinitely produces a *ray* and infinitely in both directions produces a *directed line*. This suggestion has been absorbed into mathematical physics through the concept of a Euclidean vector. [3] [4] The collection of all directed line segments is usually reduced by making "equivalent" any pair having the same length and orientation. [5] This application of an equivalence relation dates from Giusto Bellavitis’s introduction of the concept of equipollence of directed line segments in 1835.

Analogous to straight line segments above, one can also define arcs as segments of a curve.

**^**"List of Geometry and Trigonometry Symbols".*Math Vault*. 2020-04-17 . Retrieved 2020-09-01 .**^**- "Line Segment Definition - Math Open Reference".
*www.mathopenref.com*. Retrieved 2020-09-01 . **^**Harry F. Davis & Arthur David Snider (1988)*Introduction to Vector Analysis*, 5th edition, page 1, Wm. C. Brown Publishers 0-697-06814-5**^**Matiur Rahman & Isaac Mulolani (2001)*Applied Vector Analysis*, pages 9 & 10, CRC Press0-8493-1088-1**^**Eutiquio C. Young (1978)*Vector and Tensor Analysis*, pages 2 & 3, Marcel Dekker0-8247-6671-7

*This article incorporates material from Line segment on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*

## Lesson 6

This lesson is optional. In this lesson, students are introduced to how the area of a scaled copy relates to the area of the original shape. Students build on their grade 6 work with exponents to recognize that the area increases by the square of the scale factor by which the sides increased. Students will continue to work with the area of scaled shapes later in this unit and in later units in this course. Although the lesson is optional, it will be particularly helpful for students to have already had this introduction when they study the area of circles in a later unit.

In two of the activities in this lesson, students build scaled copies using pattern blocks as units of area. This work with manipulatives helps accustom students to a pattern that many find counterintuitive at first (MP8). (It is a common but false assumption that the area of scaled copies increases by the same scale factor as the sides.) After that, students calculate the area of scaled copies of parallelograms and triangles to apply the patterns they discovered in the hands-on activities (MP7).

### Learning Goals

Let's build scaled shapes and investigate their areas.

### Required Materials

### Required Preparation

Prepare to distribute the pattern blocks, at least 16 blue rhombuses, 16 green triangles, 10 red trapezoids, and 7 yellow hexagons per group of 3–4 students.

Copy and cut up the Area of Scaled Parallelograms and Triangles blackline master so each group of 2 students can get 1 of the 2 shapes.

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## 6.2: Scaling More Pattern Blocks (10 minutes)

### Optional activity

This activity extends the conceptual work of the previous one by adding a layer of complexity. Here, the original shapes are comprised of more than 1 block, so the number of blocks needed to build their scaled copies is not simply ( ext<(scale factor)>^2) , but rather (n imes ext<(scale factor)>^2) , where (n) is the number of blocks in the original shape. Students begin to think about how the scaled area relates to the original area, which is no longer 1 area unit. They notice that the pattern ( ext<(scale factor)>^2) presents itself in the factor by which the original number of blocks has changed, rather than in the total number of blocks in the copy.

As in the previous task, students observe regularity in repeated reasoning (MP8), noticing that regardless of the shapes, starting with (n) pattern blocks and scaling by (s) uses (ns^2) pattern blocks.

Also as in the previous task, the shape composed of trapezoids might be more challenging to scale than those composed of rhombuses and triangles. Prepare to support students scaling the red shape by offering some direction or additional time, if feasible.

As students work, monitor for groups who notice that the pattern of squared scale factors still occurs here, and that it is apparent if the original number of blocks is taken into account. Select them to share during class discussion.

### Launch

Keep students in the same groups, or form combined groups if there are not enough blocks. Assign one shape for each group to build (or let groups choose a shape, as long as all 3 shapes are equally represented). To build a copy of each given shape using a scale factor of 2, groups will need 12 blue rhombuses, 8 red trapezoids, or 16 green triangles. To completely build a copy of each given shape with a scale factor of 3, they would need 27 blue rhombuses, 18 red trapezoids, and 36 green triangles however, the task prompts them to stop building when they know what the answer will be.

Give students 6–7 minutes to build their shapes and complete the task. Remind them to use the same blocks as those in the original shape and to check the side lengths of each built shape to make sure they are properly scaled.

Using real pattern blocks is preferred, but the Digital Activity can replace the manipulatives if they are unavailable.

Your teacher will assign your group one of these figures, each made with original-size blocks.

Expand Image

**Description:** <p>3 shapes composed of pattern blocks. First shape, labeled figure D composed of 3 blue rhombuses. Second, labeled figure E composed of 2 red trapezoids. Third, labeled figure F composed of 4 green triangles.</p>

In the applet, move the slider to see a scaled copy of your assigned shape, using a scale factor of 2. Use the original-size blocks to build a figure to match it. How many blocks did it take?

Your classmate thinks that the scaled copies in the previous problem will each take 4 blocks to build. Do you agree or disagree? Explain you reasoning.

Move the slider to see a scaled copy of your assigned shape using a scale factor of 3. Start building a figure with the original-size blocks to match it. Stop when you can tell for sure how many blocks it would take. Record your answer.

Predict: How many blocks would it take to build scaled copies using scale factors 4, 5, and 6? Explain or show your reasoning.

How is the pattern in this activity the same as the pattern you saw in the previous activity? How is it different?

## Lines, Segments, and Rays

Although we all know intuitively what a **line** is, it is actually difficult to give a good mathematical definition. Roughly, we can say that a line is an infinitely thin, infinitely long collection of points extending in two opposite directions. When we draw lines in geometry, we use an arrow at each end to show that it extends infinitely.

A line can be named either using two points on the line (for example, A B &harr ) or simply by a letter, usually lowercase (for example, line m ).

A **line segment** has two endpoints. It contains these endpoints and all the points of the line between them. You can measure the length of a segment, but not of a line.

A segment is named by its two endpoints, for example, A B ¯ .

A **ray** is a part of a line that has one endpoint and goes on infinitely in only one direction. You cannot measure the length of a ray.

A ray is named using its endpoint first, and then any other point on the ray (for example, B A &rarr ).

## Lines, Segments, and Rays

Although we all know intuitively what a **line** is, it is actually difficult to give a good mathematical definition. Roughly, we can say that a line is an infinitely thin, infinitely long collection of points extending in two opposite directions. When we draw lines in geometry, we use an arrow at each end to show that it extends infinitely.

A line can be named either using two points on the line (for example, A B &harr ) or simply by a letter, usually lowercase (for example, line m ).

A **line segment** has two endpoints. It contains these endpoints and all the points of the line between them. You can measure the length of a segment, but not of a line.

A segment is named by its two endpoints, for example, A B ¯ .

A **ray** is a part of a line that has one endpoint and goes on infinitely in only one direction. You cannot measure the length of a ray.

A ray is named using its endpoint first, and then any other point on the ray (for example, B A &rarr ).