geometry assignment 1.1 answers

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1.1: Geometry Terms

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Define and use terms, including points, lines, planes, space, and postulates.

Geometric Definitions

A point is an exact location in space . It describes a location, but has no size. Examples are shown below:

A line is infinitely many points that extend forever in both directions. Lines have direction and location and are always straight .

Figure \(\ PageIndex{2}\)

A plane is a flat surface that contains infinitely many intersecting lines that extend forever in all directions. Think of a plane as a huge sheet of paper with no thickness that goes on forever.

We can use point , line , and plane to define new terms.

Space is the set of all points extending in three dimensions . Think back to the plane. It extended in two dimensions, what we think of as up/down and left/right. If we add a third dimension, one that is perpendicular to the other two, we arrive at three-dimensional space.

Points that lie on the same line are collinear . \(P\),\(Q\),\(R\),\(S\), and \(T\) are collinear because they are all on line \(w\). If a point U were located above or below line w, it would be non-collinear .

Points and/or lines within the same plane are coplanar . Lines h and i, and points \(A\),\(B\),\(C\),\(D\),\(G\), and \(K\) are coplanar in Plane \(J\). Line \(\overleftrightarrow{KF}\) and point \(E\) are non-coplanar with Plane \(J\).

An endpoint is a point at the end of a line segment . A line segment is a portion of a line with two endpoints. Or, it is a finite part of a line that stops at both ends. Line segments are labeled by their endpoints. Order does not matter.

A ray is a part of a line. It begins with an endpoint and extends forever away from the endpoint in one direction, perfectly straight. A ray is labeled by its endpoint and one other point on the ray. For rays, order does matter. When labeling, put the endpoint under the side WITHOUT the arrow.

An intersection is a point or set of points where lines, planes, segments, or rays overlap.

A postulate is a basic rule of geometry. Postulates are assumed to be true (rather than proven), much like definitions. The following is a list of some basic postulates.

Postulate #1: Given any two distinct points, there is exactly one (straight) line containing those two points.

Postulate #2: Given any three non-collinear points, there is exactly one plane containing those three points.

Postulate #3: If a line and a plane share two points, then the entire line lies within the plane.

Postulate #4: If two distinct lines intersect, the intersection will be one point.

Lines \(I\) and \(m\) intersect at point \(A\).

Postulate #5: If two distinct planes intersect, the intersection will be a line.

Figure \(\PageIndex{13}\)

When making geometric drawings, be sure to be clear and label all points and lines.

What if you were given a picture of a figure or object, like a map with cities and roads marked on it? How could you explain that picture geometrically?

Example \(\PageIndex{1}\)

What best describes San Diego, California on a globe: point, line, or plane?

A city is usually labeled with a dot, or point, on a globe.

Example \(\PageIndex{2}\)

Use the picture below to answer these questions.

Figure \(\ PageIndex {14}\)

• List another way to label Plane \(J\).
• List another way to label line \(h\).
• Are \(K\) and \(F\) collinear?
• Are \(E\),\(B\) and \(F\) coplanar?
• Plane \(BDG\) is one possibility. Any combination of three coplanar points that are not collinear would be correct.
• \(\overleftrightarrow{AB}\). Any combination of two of the letters \(A\), \(B\), or \(C\) would also be correct.
• Yes, they both lie on \(\overleftrightarrow{KF}\).
• Yes, even though \(E\) is not in Plane \(J\), any three points make a distinct plane. Therefore, the three points create Plane \(EBF\).

Example \(\PageIndex{3}\)

What best describes a straight road that begins in one city and stops in a second city: ray, line, segment, or plane?

The straight road connects two cities, which are like endpoints. The best term is segment.

Example \(\PageIndex{4}\)

Answer the following questions about the picture.

• Is line \(l\) coplanar with Plane \(V\), Plane \(W\), both, or neither?
• Are \(R\) and \(Q\) collinear?
• What point belongs to neither Plane V nor Plane \(W\)?
• List three points in Plane \(W\).
• Any combination of \(P\), \(O\), \(T\), and \(Q\) would work.

Example \(\PageIndex{5}\)

Draw and label a figure matching the following description: Line \(\overleftrightarrow{AB}\) and ray \(\overrightarrow{CD}\) intersect at point \(C)\. Then, redraw so that the figure looks different but is still true to the description.

Neither the position of A or B on the line, nor the direction that \(\overrightarrow{CD}\) points matter.

For the second part, this is one way to draw the diagram differently:

For questions 1-5, draw and label a figure to fit the descriptions.

• \(\overrightarrow{CD}\) intersecting \(\overline{AB}\) and Plane \(P\) containing \(\overline{AB}\) but not \(\overrightarrow{CD}\).
• Three collinear points \(A\), \(B\), and \(C\). \(B\) is also collinear with points \(D\) and \(E\).
• \(\overrightarrow{XY}\), \(\overrightarrow{XZ}\), and \(\overrightarrow{XW}\), such that \(\overrightarrow{XY}\) and \(\overrightarrow{XZ}\) are coplanar, but \(\overrightarrow{XW}\) is non-coplanar with both of the other rays.
• Two intersecting planes, \(P\) and \(Q\), with \(\overline{GH}\), where \(G\) is in plane \(P\) and \(H\) is in plane \(Q\).
• Four non-collinear points \(I\), \(J\), \(K\),and \(L\), with line segments connecting all points to each other.
• Name this line in five ways.

Figure \(\PageIndex{18}\)

• Name the geometric figure in three different ways.

Figure \(\PageIndex{19}\)

• Name the geometric figure in two different ways.

Figure \(\PageIndex{20}\)

• What is the best possible geometric model for a soccer field? Explain your answer.
• List two examples of where you see rays in real life.
• What type of geometric object is the intersection of a line and a plane? Draw your answer.
• What is the difference between a postulate and a theorem ?

For 13-16, use geometric notation to explain each picture in as much detail as possible.

Figure \(\PageIndex{23}\)

For 17-25, determine if the following statements are true or false.

• Any two points are collinear.
• Any three points determine a plane.
• A line is two rays with a common endpoint.
• A line segment is infinitely many points between two endpoints.
• A point takes up space.
• A line is one-dimensional.
• Any four points are coplanar.
• \(\overrightarrow{AB}\) could be read “ray \(AB\)” or “ray \(BA\).”
• \(\overleftrightarrow{AB}\) could be read “line \(AB\)” or “line \(BA\).”

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.1.

Additional Resources

Interactive Element

Video: Basic Geometric Definitions Principles - Basic

Activities: Basic Geometric Definitions Discussion Questions

Study Aids: Basics of Geometry Study Guide

Practice: Geometry Terms

Real World: Basic Geometric Definitions

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1.1.4: Geometry

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• David Lippman
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Geometric shapes, as well as area and volumes, can often be important in problem solving.

You are curious how tall a tree is, but don’t have any way to climb it. Describe a method for determining the height.

There are several approaches we could take. We’ll use one based on triangles, which requires that it’s a sunny day. Suppose the tree is casting a shadow, say 15 ft long. I can then have a friend help me measure my own shadow. Suppose I am 6 ft tall, and cast a 1.5 ft shadow. Since the triangle formed by the tree and its shadow has the same angles as the triangle formed by me and my shadow, these triangles are called similar triangles and their sides will scale proportionally. In other words, the ratio of height to width will be the same in both triangles. Using this, we can find the height of the tree, which we’ll denote by \(h\):

\[\frac{6 \mathrm{ft} \text { tall }}{1.5 \mathrm{ft} \text { shadow }}=\frac{h \mathrm{ft} \text { tall }}{15 \mathrm{ft} \text { shadow }}\nonumber \]

Multiplying both sides by 15, we get \(h = 60\). The tree is about 60 ft tall.

It may be helpful to recall some formulas for areas and volumes of a few basic shapes.

Area: \(L \cdot W\)

Perimeter: \(2 L+2 W\)

Circle , radius \(r\)

Area: \(\pi r^{2}\)

Circumference = \(2 \pi r\)

Rectangular Box

Volume: \(L \cdot W \cdot H\)

Volume: \(\pi r^{2} H\)

If a 12 inch diameter pizza requires 10 ounces of dough, how much dough is needed for a 16 inch pizza?

To answer this question, we need to consider how the weight of the dough will scale. The weight will be based on the volume of the dough. However, since both pizzas will be about the same thickness, the weight will scale with the area of the top of the pizza. We can find the area of each pizza using the formula for area of a circle, \(A=\pi r^{2}\):

A 12” pizza has radius 6 inches, so the area will be \(\pi 6^{2}\) = about 113 square inches.

A 16” pizza has radius 8 inches, so the area will be \(\pi 8^{2}\) = about 201 square inches.

Notice that if both pizzas were 1 inch thick, the volumes would be 113 in 3 and 201 in 3 respectively, which are at the same ratio as the areas. As mentioned earlier, since the thickness is the same for both pizzas, we can safely ignore it.

We can now set up a proportion to find the weight of the dough for a 16” pizza:

\(\frac{10 \text { ounces }}{113 \mathrm{in}^{2}}=\frac{x \text { ounces }}{201 \mathrm{in}^{2}}\) Multiply both sides by 201

\(x=201 \cdot \frac{10}{113}=\) about 17.8 ounces of dough for a 16” pizza.

It is interesting to note that while the diameter is \(\frac{16}{12}=1.33\) times larger, the dough required, which scales with area, is \(1.33^{2}=1.78\) times larger.

We would expect the calories to scale with volume. Since the marshmallows have cylindrical shapes, we can use that formula to find the volume. From the grid in the image, we can estimate the radius and height of each marshmallow.

The regular marshmallow appears to have a diameter of about 3.5 units, giving a radius of 1.75 units, and a height of about 3.5 units. The volume is about \(\pi(1.75)^{2}(3.5)=33.7 \text { units}^{3}\).

The jumbo marshmallow appears to have a diameter of about 5.5 units, giving a radius of 2.75 units, and a height of about 5 units. The volume is about \(\pi(2.75)^{2}(5)=118.8 \text { units}^{3}\).

We could now set up a proportion, or use rates. The regular marshmallow has 25 calories for 33.7 cubic units of volume. The jumbo marshmallow will have:

\(118.8 \text { units }^{3} \cdot \frac{25 \text { calories }}{33.7 \text { units }^{3}}=88.1 \text { calories }\)

It is interesting to note that while the diameter and height are about 1.5 times larger for the jumbo marshmallow, the volume and calories are about \(1.5^{3}=3.375\) times larger.

Try it Now 5

A website says that you’ll need 48 fifty-pound bags of sand to fill a sandbox that measure 8ft by 8ft by 1ft. How many bags would you need for a sandbox 6ft by 4ft by 1ft?

The original sandbox has volume \(64 \mathrm{ft}^{3}\). The smaller sandbox has volume \(24 \mathrm{ft}^{3}\).

\(\frac{48 \mathrm{bags}}{64 \mathrm{ft}^{3}}=\frac{x \mathrm{bags}}{24 \mathrm{ft}^{3}}\) results in \(x= 18\) bags.

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