Mathematics: Levels E, M, D, and A Study Guide for the TABE Test

Geometry: Part 2

Percentage of Test Level Specifically Assessing Geometry (— = Assumed)

L	E	M	D	A
11%	10%	10%	18%	15%

Circles and 3-D Shapes Containing Circles

In geometry, a circle is defined as the set of all the points that are in the same plane and are the same distance from a center point. So every part of this circle is the same distance from the center point \(O\).

• The distance from \(O\) to \(B\) is the same as the distance from \(O\) to \(Y\) and \(O\) to \(Z\): \(OB = OY = OZ\).

• Each one of the segments \(\overline{OB}\), \(\overline{OY}\), and \(\overline{OZ}\) is a radius (the plural is radii) of the circle.

• The segment \(\overline{BZ}\) goes through the center and is a diameter. A diameter is twice as long as a radius.

The distance around the circle is called the circumference, and it can be calculated using either \(C = \pi d\) or \(C=2\pi r\), where \(d\) is the diameter length, \(r\) is the radius length, and \(\pi\) is pi, the number \(3.14\).

The circle can be divided into \(360\) little pieces called degrees, written \(360^\circ\). A half-circle (semicircle) is \(180^\circ\), \(90^\circ\) would be a quarter circle, and so on.

Any portion of a circle is called an arc. For example, the part from \(Y\) to \(Z\) is called arc \(YZ\), and its symbol is \(\overset{\frown}{YZ}\).

If you know the number of degrees in an arc and you need to calculate its length, you can use this formula: \(L= \frac{m}{360} \cdot 2 \pi r\). The variable \(m\) is the measure of the arc in degrees. Here’s an example using the figure above.

Let’s say that the arc \(\overset{\frown}{YZ}\) has a measure of \(30^\circ\) and radius \(\overline{OY} = 12\) inches. We want to know how long the arc is.

\[L= \frac{m}{360} \cdot 2 \pi r\] \[L=\frac{30}{360} \cdot 2 \cdot 3.14 \cdot 12\] \[L= \frac{1}{12} \cdot 75.36\] \[L = 6.28 \text{ inches}\]

The area of a circle is calculated using \(A = \pi r^2\).

Points and Lines

In geometry, lines and points have a narrower meaning than we use in everyday life. We’ll look at the way these terms are used in the next few sections.

Definitions

When we think of a point, we probably all picture some kind of little spot or dot, and that idea works pretty well in geometry too. Technically, a point is an infinitely small location, but we can’t actually put that on paper, so we just put a small dot down and call it a point. Points are named with a capital letter.

The idea of a line in geometry is something that is straight, infinitely thin, and infinitely long. Being infinitely thin, we can’t see a true line, so we just draw something straight and say, “The line is in there”. The arrowheads at either end of a line show that it continues forever in both directions. A line has no endpoints. Lines are usually named using two points that are on the line. The line below would be named line \(AB\) and the symbol for it is \(\overleftrightarrow{AB}\).

A line segment is a piece of a line and it has two endpoints. The piece of the line above between \(A\) and \(B\) would be named segment \(AB\), and the symbol for it is \(\overline{AB}\).

A ray is kind of like half a line. It has one endpoint and continues infinitely far in one direction, a lot like a laser. A ray is named using two points, always starting with the endpoint. The ray below would be named ray \(AB\), and its symbol is \(\overrightarrow{AB}\).

Perpendicular and Parallel Lines

Perpendicular lines (or segments) meet to form four right angles (\(90^\circ\)). Below is an example:

Line \(AB\) is perpendicular to line \(CD\). Using symbols, this would be written \(\overleftrightarrow{AB} \perp \overleftrightarrow{CD}\). The little square where the lines meet is the symbol for a right angle. Although not specifically shown, the other three angles there are right angles too.

Parallel lines are lines in the same plane that never meet, no matter how far they are extended. They are drawn like this:

We could say that line \(AB\) is parallel to line \(CD\), or, in symbols, we could write \(\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}\).

Angles

An angle is formed when two rays or line segments share a common endpoint. Angles are named using the angle symbol, three letters, one from each side and the vertex point. The vertex point is always in the middle, but it doesn’t matter which order the other two points are in. The right angle shown below could be named \(\angle CAR\) or \(\angle RAC\).

Types of Angles

Right angles have measures of \(90^\circ\). Acute angles are less than \(90^\circ\), and obtuse angles are greater than \(90^\circ\).

Angles in a Figure

In the two figures below you can see examples of adjacent angles. They share a vertex and a common side between them.

Two angles that add up to \(90^\circ\) are said to be complementary, so \(\angle ABD\) and \(\angle DBC\) in the figure above are a pair of complementary angles.

Two angles that add up to \(180^\circ\) are said to be supplementary angles, so in the figure above, \(\angle CAT\) and \(\angle TAR\) are a pair of supplementary angles.

An angle can also be named by putting a number between the two sides near the vertex, as above. Keep in mind that these are not degrees.

Also in the above figure you can see that when two lines intersect they form four angles. The pairs of angles that are opposite from each other are called vertical angles (in this case, vertical doesn’t mean straight up). So, \(\angle 1\) and \(\angle 3\) are a pair of vertical angles and so are \(\angle 2\) and \(\angle 4\). Important fact: vertical angles are always congruent (they measure the same) to each other. We could write \(\angle 1 \cong \angle 3\) and \(\angle 2 \cong \angle 4\).

Keeping in mind these properties, you should be able use them to solve geometric problems. Let’s look at the last figure again.

Suppose \(\angle 1 = 60^\circ\). What is the sum of \(\angle 2\) and \(\angle 4\)?

How to solve it:

\(\angle 1\) and \(\angle 2\) are supplementary, so \(\angle 1 + \angle 2 = 180^\circ\).

\(\angle 1\) is \(60^\circ\) so \(\angle 2\) must be \(120^\circ\) to make a sum of \(180^\circ\).

\(\angle 2\) and \(\angle 4\) are vertical angles, so they are congruent and \(\angle 2 = \angle 4\).

Therefore, \(\angle 4=120^\circ\).

\(\angle 2 = 120^\circ\) and \(\angle 4 = 120^\circ\), so their sum is \(240^\circ\).

The Coordinate Grid

You may know about latitude and longitude, the system used to locate points on planet Earth. It uses numbered north-south lines and east-west lines. By stating which lines cross each other at a certain spot, we can identify any location on Earth. This organization of Earth’s surface is called a coordinate system.

On a much smaller scale, we have a coordinate system in math to locate points on a flat surface. It uses two number lines that cross at their zero points. We call those the axes (that’s “AX-ees”, not the tools you chop trees down with). The horizontal axis is called the \(x\)-axis, and the vertical axis is called the \(y\)-axis.

To locate a point using the \(x-y\) coordinates, we use a pair of numbers inside parentheses called an ordered pair. The point where the axes cross is the point \((0, 0)\). Look at point \(A\). It has the coordinates \((4, 5)\). The first number tells us that the point is \(4\) units to the right of \((0, 0)\) in the \(x\) direction. The \(4\) is called the \(x\)-coordinate. It always comes first.

After moving \(4\) units to the right, the second number tells you that point \(A\) is \(5\) units up, in the \(y\) direction. The \(5\) is called the \(y\)-coordinate. Together, they make up an ordered pair \((x, y)\). In the case of \(A\), \(x = 4\) and \(y = 5\).

Something you may notice if you look at the numbers along the \(x\) and \(y\) axes is that they get bigger going to the right or up from \((0,0)\). Also, they get bigger negatively as they go to the left or down. (Technically they are getting smaller, since \(-3\), for example, is lower than \(-2\).)

Look at point \(B\). The directions to get there from \((0, 0)\) can be written this way: go left to \(4\) and go up to \(5\). As a shortcut, we can write \((-4,5)\) because \(-4\) means go left and positive \(5\) means go up.

Finally, look at point \(C\). To get there from \((0, 0)\) we would need to go left to \(-4\), and down to \(-5\). The \((x, y\)) coordinates of point \(C\) would be \((-4, -5)\).

Quadrants

In the \(x-y\) system, the axes make four sections that points can lie in. These sections are called quadrants and they are numbered as shown below.

Scale Drawings

Be able to solve problems using scale drawings to find lengths or areas of real objects, like the problem below.

Adapted from: https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications

On the map above, Seattle, Portland, and Boise form a triangle with the dimensions shown. If the actual distance from Seattle to Boise is \(400\) miles, how far is it from Seattle to Portland?

To do this kind of problem we need to set up a proportion between the map distances and the actual distances. We will write two ratios with the map values on top and the real distances on the bottom. Not knowing the real distance from Seattle to Portland, we write \(x\) in its place.

\[\frac{4 \text{ in}}{400 \text{ mi}} = \frac{1.5 \text{ in}}{x}\]

Ignore the labels for clarity and cross-multiply:

\[x \cdot 4 = 400 \cdot 1.5\] \[4x=600\]

Dividing both sides by \(4\), we get:

\[x=150\]

It’s therefore \(150\) miles from Seattle to Portland.

In the drawing above are two scale drawings of a wedge. (Remember that “ can be used to represent “inches” and ‘ can be used to stand for “feet”.) If the scales used are \(1” = 3’\) and \(1”= 6’\), which triangle has the scale \(1”=6’\) ? The answer is \(\triangle DOG\) and here’s why: Using a scale of \(1” = 3’\), it would take \(2”\) to represent \(6’\). That’s twice as long as in the scale \(1” = 6’\), so that would go along with the bigger triangle, \(\triangle CAT\).

If that’s confusing, imagine a drawing with a scale of \(1”=500’\). It would take only a tiny fraction of an inch to represent \(6’\), and you’d have a very tiny triangle.

Density

Density is a word that loosely means how tightly packed something is. In school, teachers are concerned with how many students are packed into each classroom. While \(20\) students per classroom is a fairly low density, \(40\) is very high. City dwellers live where there is a high population density, measured in people per square mile. Gold has a very high weight density, which could be measured in pounds per cubic foot. (Side note: A cubic foot of gold weighs more than half a ton.)

The point of this section is that there are different kinds of densities. What they have in common is that they all involve some ratio where the bottom number is one, and they use the word per in their description. You should be able to calculate a density given two values as shown below.

If \(3\) cubic feet of maple wood weighs \(120\) pounds, what is its density in pounds per cubic foot? Whatever unit comes before the word per goes on top of the ratio and the other unit goes on the bottom. Here we have \(\frac{120\ lb}{3\ cu ft}\). When we simplify that, it comes to \(40 \frac{lb}{cu ft}\).

The Pythagorean Theorem

The Pythagorean theorem is a pattern that applies to all right triangles. To work with it you need to know what a square root is, and it really helps if you know the squares of the following numbers:

\[\begin{array}{cccc} &2^2=4&5^2=25&8^2=64&11^2=121\\ &3^2=9&6^2=36&9^2=81&12^2=144\\ &4^2=16 &7^2 =49&10^2 =100\\ \end{array}\]

In the right triangle above, sides \(a\) and \(b\) are the shorter sides and are called the legs of the triangle. The longest side, \(c\), is called the hypotenuse. The Pythagorean theorem states that the sum of the squares of the two legs equals the square of the hypotenuse. In terms of \(a, b,\) and \(c\), the theorem is written as such:

\[a^2 + b^2 = c^2\]

\(a\) and \(b\) are always the shortest two sides, the legs, of the right triangle, and \(c\) is always the longest side, the hypotenuse. This lets us figure out the length of any side if we know the other two sides. Here’s an example:

How long is side x in this triangle?

We can let \(a=6, b=8,\ \ \ \text{ and } c=x\).

So:

\[6^2 + 8^2 = x^2\] \[36 + 64 = x^2\] \[100 = x^2\] \[\sqrt{100} = \sqrt{x^2}\] \[10 = x\]

Another example comes from the building trades. When building a house, carpenters measure the diagonals of the foundation and could use the Pythagorean theorem to make sure all the corners are right angles.

Suppose the drawing below represents the layout of a small tool shed. Given the dimensions shown, will the shed be a true rectangle? Remember, you can’t go by looks, only by the given numbers.

We just need to see if the dimensions are in line with the Pythagorean theorem.

Is \(9^2 + 12^2 = 14^2\) true? We’ll do the math and figure out:

\[81 + 144 = 196\] \[225 \neq 196\]

It is not true, so these dimensions don’t make a true rectangle.

Note: Carpenters don’t actually do this. Instead, they just measure both diagonals. Knowing the sides have the correct dimensions, if the diagonals are equal, the building will be a true rectangle. Still, it makes a good problem for us.

The Pythagorean theorem is sometimes used in an \(x-y\) coordinate system.

Suppose one point is at \((2, 5)\) and another point is at \((5, 1)\). How far apart are the points?

If you sketch a segment from \(A\) to \(B\), you can see that we can draw a right triangle with \(\overline {AB}\) as a hypotenuse and legs with lengths \(3\) and \(4\). The Pythagorean theorem tells us that \(a^2+b^2=c^2\), so let \(a=3, b=4, \overline{AB} = c,\) and calculate the length of \(c\):

\[3^2+4^2=c^2\] \[9+16=c^2\] \[25 = c^2\] \[\sqrt{25} = \sqrt{c^2}\] \[5 = c\]