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Wize High School Geometry Textbook (Common Core) > Foundations of Geometry

Measuring and Constructions

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Measuring


Up to this point we've been making all kinds of objects, but we haven't mentioned how big these are.
A line segment could be small like 1cm or large like 35 km! To make better sense of this, we'll need to start measuring things.

Distance

Points on a line can be paired up with the real line. In this way we can assign a number to any point on a line. This is called the coordinate of a point.

To measure the distance between two points:
  1. Use the real line to assign each point a coordinate
  2. Subtract one coordinate from the other
  3. Take the absolute value of the result
Example 1
What is the distance between points XX and YY?
ANSWER:
7−4=3∣3∣=37 - 4 = 3 \\ |3| = 3

The distance is 3

Wize Tip
A ruler can be used as a real line so that we can easily assign coordinates to our points.

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Coordinate Axis

All of the objects we've discussed can be drawn on a plane. We will call this plane the coordinate axis.
It is a plane with two lines that cross at right angles called the x-axis and the y-axis.

The point where they intersect the origin.
With this we can then measure the location of any point drawn on a coordinate axis. The coordinate of a point on the coordinate axis is an ordered pair of numbers.
(x,y)(x,y)
The first number represents the distance in the x direction (from the point to the y-axis), and the second number represents the distance in the y direction (from the point to the x-axis).

Congruent Segments

We say that two line segments are congruent if they have exactly the same length.
For notation we will use AB‾≅CD‾\overline{AB} \cong \overline{CD} to show that AB‾\overline{AB} and CD‾\overline{CD} are congruent.

Example 2
  1. Draw a point at (−1,3)(-1, 3) labeled PP.
  2. Draw a point at (4,3)(4, 3) labeled RR .
  3. Draw a point at (−1,−1)(-1, -1) labeled QQ.
  4. Draw the line segments PR‾\overline{PR} and PQ‾\overline{PQ}.
What is the length of the segment PR‾\overline{PR}?

ANSWER: The length of PR‾=5\overline{PR} = 5

Are PR‾\overline{PR} and PQ‾\overline{PQ} congruent?

ANSWER: The length of PQ‾=4\overline{PQ} = 4.
Since the lengths are not the same, these segments are not congruent.
PR‾≆PQ‾\overline{PR} \ncong \overline{PQ}

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Basic Constructions


Many of the objects in geometry can be built, or constructed, using simple tools. This process is important for not only visualizing objects, but also for learning how they are connected.

Tools

A pencil is a common tool for drawing points and various other objects in geometry. Any thing that leaves a mark, such as a pen, will also work. To make the the constructions precise, we will use a marking tool along with a straightedge and compass.
A straightedge is any object with a straight edge on it like a ruler. It is used to draw lines, segments, or rays.
A compass has one drawing end and one fixed end. It can be used to mark off distances, or used to sweep out arcs.

Technology

There are many programs available that allow you to digitally make constructions. One advantage of these programs is they allow you to move points without changing the relationships to each other.


Example
  1. Make two separate line segments PQ‾\overline{PQ} and RS‾\overline{RS}.
  2. Set the compass so that it touches both point PP and QQ.
  3. Use the compass to sweep out a circle around point PP .
ANSWER:
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Constructing congruent segments

Our goal in this construction is create new segment that is congruent to a given one.

Using a compass and straightedge

Start with a segment AB‾\overline{AB}
  1. Draw a line segment slightly larger than the given segment.
  2. Label one end of this segment as CC .
  3. Adjust the compass width to be the same as AB‾\overline{AB}
  4. Set the fixed end at point CC, and make a mark on your new segment.
  5. Label the intersection as point DD.
From this we now have AB‾≅CD‾\overline{AB} \cong \overline{CD}


Using Technology


Begin with a segment AB‾\overline{AB}
  1. Draw a line segment slightly larger than the given segment.
  2. Label one end of this segment as CC.
  3. Use the compass tool to create a circle with the radius the same as AB‾\overline{AB}.
  4. Now place the circle from the compass so that the center is at point CC.
  5. Place a point DD at the intersection of the segment and the circle.
From this we now haveAB‾≅CD‾\overline{AB} \cong \overline{CD}
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Road Trip



You and a friend have decided to take a road trip. Use the diagram below and the given information to answer the questions.


According to the map you will need to travel a total distance of 412 miles traveling along the highway if you start at city B and end city E.

1. If the distance between city B and city C is 98 miles and the distance between city C and city D is 75 miles, what is the distance between city D and city E?

ANSWER: The distance is 239 miles.

BC‾+CD‾+DE‾=41298+75+DE‾=412173+DE‾=412DE‾=239\begin{aligned} \overline{BC} + \overline{CD} + \overline{DE} &= 412 \\ 98 + 75 + \overline{DE} & = 412 \\ 173 + \overline{DE} &= 412 \\ \overline{DE} &= 239 \end{aligned}

2. Your friend takes out a ruler and measures the distance between cities B and E. With the help of the map key, they find that this distance is only 119 miles. Why is this so different from the amount of distance you will actual end up travelling?

ANSWER: The distance of 119 miles is measured is in a strait line between the two cities. In your trip, however, you will be traveling along the roads which will first take you to city C, then city D, and lastly city E. These are not in a strait line, so each of their distances are added separately to give a total of 412 miles.

Lengths of segments




Determine if segments PQ‾\overline{PQ} and RS‾\overline{RS} are congruent for each.

  1. P(1,3),Q(1,6),R(−2,−2),S(−2,1)P(1, 3), Q(1,6), R(-2, -2), S(-2, 1)
  2. P(−1,−3),Q(−1,4),R(2,−3),S(2,3)P(-1, -3), Q(-1, 4), R(2, -3), S(2, 3)
  3. P(0,0),Q(0,6),R(2,3),S(−2,3)P(0,0), Q(0, 6), R(2, 3), S(-2, 3)
Determine if the segments PQ‾\overline{PQ} and RS‾\overline{RS} are congruent.

1. P(1,3),Q(1,6),R(−2,−2),S(−2,1)P(1, 3), Q(1,6), R(-2, -2), S(-2, 1)

Segments and Algebra

Use the diagram below, and the fact that the length of AC‾\overline{AC} is exactly 23 cm to answer the questions.

1. What is the value of x ?

2. What is the length of AB‾\overline{AB}?

3. What is the length of BC‾\overline{BC}?

1. What is the value of x ?

Garden Measuring



While gardening Mr. Jones noticed he had arranged his plants according to the diagram.


The measured distance from AA to DD was exactly 36 feet.
He also found that AB‾≅BC‾\overline{AB} \cong \overline{BC}, and that AC‾≅CD‾\overline{AC} \cong \overline{CD}.

Find the distance between all of the plants.
The distance from AA to BB is how many feet?