### Archive for the ‘Uncategorized’ Category

For reasons known only to ETS, number line problems are categorized with geometry. Such problems are basically algebra problems: they present you with some diagram or some facts about certain variables and then ask whether some equations or inequalities could be true, given certain relationships among the variables. Here is a straightforward example:

__Example 1
__

__x__

__y__

A. is greater than

B. is greater than

C. The two quantities are equal

D. It cannot be determined which, if any, is greater.

Now, a number line simply gives all the numbers from least to greatest. So the numbers to the right of the number line are greater than those to the left. And if there are markings on the number line, then you may (provided the question says nothing to the contrary) assume that the markings are evenly spaced. What does this mean? Well in the following diagram:

you know that the distance between and is only half of the distance between and . Using algebra, we could express this as:

Now, how do we solve number line problems? It will depend on what the question asks for. Some questions ask for which of the following *could* be true, whereas others ask about which of the following *must* be true.

If the questions asks which equations could be true, then we are looking for either a set of numbers that makes the equation true and is compatible with the diagram/given information, or we want to show that the equation is somehow inconsistent with our diagram/given information.

If the questions asks which equations must be true, then we are looking for either a set of numbers that is compatible with the diagram/given information but makes our equation false, or we want to show that the equation somehow follows from the diagram/given information.

Whether you should look for a specific example or some proof of the consistency/inconsistency of the given equation is a judgment call that you need to make in the moment. It's hard to give general principles about when to do which, but after doing some practice problems and just thinking about the equation at hand, you should develop some sense of how to decide if an equation is consistent or not. For example:

__Example 2__

Let and . Must the following be true?

After doing some practice problems, you get a sense of which numbers to consider when you come to one of these problems. And by running through those numbers quickly, you can often show that some equation either can be satisfied (which answers whether the equation could be true) or can be violated (which answers whether the equation must be true).

The only real piece of advice I have here is two-fold: First, always try easy, concrete numbers. Don't just think about "one negative and one positive value." It's a lot more general to think that way, but also a lot harder to evaluate. Try, instead, or and . Nice, easy numbers to calculate. And if you want a small number, try or Second, it is easy to think about what happens if both are normal, natural numbers (e.g. 1 and 2). But also think about what happens if they are small, positive numbers (e.g. and ) or only one is positive (e.g. 1 and -1) or both are negative (e.g. -1 and -2). If the equation works (or is always violated) in all of those cases, then probably it's always true (or always false).

When it comes to proving that some equation is either always true/false given the diagram/information in the question, look for easy inferences. And if you can't see a way to prove it quickly, consider just flagging it and coming back. If you've gone through a few diverse examples and they all come out the same way, probably the equation actually is always true (or always false) and the time spent confirming that might be better spent on other problems.

__Practice Problems:__

1. The markings on the below number line are evenly spaced:

Which of the following must be true (select all that apply):

A.

B.

C.

2. Suppose we know and Which of the following must be true (select all that apply):

A.

B.

C.

3. Suppose we know Which of the following must be true (select all that apply):

A.

B.

C.

D.

E.

4. The markings in the below number line are equally spaced. Which of the following could be true:

A.

B.

C.

5. The markings in the below number line are equally spaced. Which of the following must be true:

A.

B.

C.

Geometry questions sometimes involve various graphs. For example:

__Example 1__

Find the area of the following square:

Now, these problems do not differ substantially from what we have before. Essentially, graphing problems give you the same information in a slightly different way. So instead of telling you how long the sides of a particular shape are, they might give you the coordinates of the endpoints. But of course, we know how to infer the length of that side, for we know the Pythagorean Theorem:

__Example 2__

A rectangle has a side whose vertices are at (1, 3) and (2, 2). Find the length of that side.

In this vein, a more realistic problem might look like:

__Example 3__

A triangle has vertices at (1, 4), (2,3), and (0,0). Find its perimeter.

Similarly, graphing problems can involve angles:

__Example 4__

A triangle has vertices at and . Find the degree of angle .

In short, if you have a geometry problem that involves certain points or a diagram, you can generally use those points or that diagram to infer aspects like the lengths of certain sides or the degrees of various angles, and then use that information as you normally would.

**Practice Problems:**

1. A rectangle (shown below) has vertices at (0,1), (2, 0), and (2,5). Where is its fourth vertex?

2. A circle has its center at point and its circumference includes the point . Find its radius.

3. A triangle has vertices at (1, 4), (2,3), and (0,0). Find its area.

4. The square in the following diagram has points at (2, 1) and (1, 4) as labelled. What is the area of the square?

There are also two relationships between angles and circles which are crucial. First, let's define the term "central angle":

** Central Angle:** A central angle is an angle located at the center of a circle with endpoints on the circumference of that circle

Now, the crucial fact to learn is:

** Central Angles and Arcs:** In the following diagram:

The degree of angle AOB =

Thus, we can determine the degree of any central angle by looking at the length of its arc, and vice versa.

**Example 1**

The circle below has a radius of 2 and has a length of . What is the degree of angle ?

__Example 2__

The circle below has a radius of 6 and the degree of is 47 degrees. What is the length of ?

A related fact to the above is:

** Circumference Angle Theorem:** In the following circle with radius :

The degree of angle ACB = .

This theorem is also called the Inscribed Angle Theorem.

In essence, if you have an angle on the circumference of a circle, its degree will be one-half the degree of a central angle with the same endpoints. In pictures:

Now, what may be surprising about this is that it does matter where on the circumference the angle begins from. In other words:

the angles and above have the same degree.

Finally, you must be careful to make sure that you are comparing the angle on the circumference to the correct central angle. For example:

and do not have the same degree. That is because they give rise to different central angles:

Thus, it is not enough to look at the endpoints of your angle. You also need to look at where on the circumference the angle is, so you can be sure to compare it to the relevant central angle.

__Practice Problems__

1. The below circle has a center at . Find :

2. Find the value of

3. In the following diagram, Find

4. The radius of the following circle is Find

5. The radius of the following circle is 1. Find

6. The radius of the below circle is 1. Find

7. In the below circle, which is larger?

__ __

A.

B.

C. They are equal.

D. It cannot be determined.

Now, with respect to triangles, there are two main facts you need to know:

** Triangles have 180 degrees:** The angles of a triangle add up to 180 degrees.

** Reverse Pythagorean Theorem:** Triangles whose side lengths obey the Pythagorean Theorem (i.e. the triangle has side lengths and ) must have a right angle.

Sometimes, a problem will be somewhat sneaky: instead of directly telling you that a triangle is a right triangle, it will give you the side lengths and leave you to make that inference. (Recall here our Reverse Pythagorean Theorem)

The final fact which can help in finding an unknown angle is the following formula for the sum of all the angles of a polygon:

** Polygon Angle Sum Formula:** For an -sided 2D shape, the sum of all the interior angles is equal to

What is an interior angle? Well on a polygon, we have some sides and some vertices (points where the sides meet). For example:

At each vertex, there is are two angles:

The interior angle, sensibly enough, is the angle that is insidethe polygon. In the above diagram, it is the blue angle.

Thus, our formula lets us calculate, for any polygon, the sum of all of its interior angles. This can be helpful when you are given almost all of the angles of a polygon and you need to find the final angle.

__Practice Problems
__

1. What is the sum of the interior angles for a 13-sided polygon?

2. Find the value of any interior angle of a regular 14-sided polygon.

3. Find the angles of the below polygon:

Given parallel lines, there are four main relationships that angles can have to one another: vertical angles, interior angles relationships, exterior angle relationships, and corresponding angles. These relationships can all be derived from one another, and in what follows, we include some of those derivations so that, if you forget one of them on exam day, you can recover it from knowledge of the others.

** Vertical Angles:** Two angles formed by the same lines are vertical angles if they are on opposite sides of the point of intersection between the two lines.

This is a fundamental fact that is often critical in finding an unknown angle. For example:

**Example 1:**

__Interior Angles:__

In the above diagram,

**Example 2:**

In the following diagram, let and let What is the value of ?

__Exterior Angles:__

In the above diagram,

**Example 3:**

In the below diagram, let and What is the value of ?

__Corresponding Angles:__ Two angles (which are in the same relative position) formed by a single line cutting across two parallel lines. Such angles are equal.

For example:

In this diagram, and are corresponding angles and thus equal in degree.

__Practice Problems:
__

1. Find the value of .

2. Find the value of .

3. Find the value of

4. Find the value of

Many geometry problems either directly ask about the degree of certain angles or will require you to figure out the degree of an angle in the course of solving some problem. In successive posts, we will cover the basics of what an angle is, how we can use parallel lines to find unknown angles, and how we can use triangles/circles/polygons to find unknown angles.

__Angles: Some Preliminaries__

To understand angles, we need to begin with the idea of a line:

** Line: **a straight line that continues in both directions without end.

Now, an angle appears where two lines intersect. More formally,

** Angle:** a measure of how much you would need to turn one line to make them part of the same line

For example,

And as you can see in the above diagram, there are actually four angles formed when two lines intersect. So there is a convention for naming each of the four angles:

[Practice problems with matching angles: label points on the lines and degree measures and have them match the two]

Now, here are two important facts about angles:

** Lines Have 180 Degrees:** The angle formed by a single line has 180 degrees.

** Circles Have 360 Degrees:** The angle formed by a circle is 360 degrees.

Now, we also have two relationships among lines which are very important:

** Parallel Lines:** Parallel lines go in the same direction and therefore never intersect. We often denote that lines and are parallel by writing , as in the below diagram:

__ Perpendicular Lines:__ Perpendicular lines intersect at a 90 degree angle. We denote two lines as perpendicular by writing :

In our next post, we will talk about some of the relationships that exist between angles and lines and shapes.

Below, we include some additional geometry practice problems.

1. What is the area of ? Of ?

2. In the following diagram, What is the area of ?

3. What is the area of ?

4. The radius of the below circle is 3. Find the area of the shaded region.

5. The circle below has a radius of . Find the ratio of the shaded region's area to the area of the square below.

The next 3D shape we shall look at is the circular cylinder (also called the “right circular cylinder” by the GRE for its right angles):

__Circular Cylinder:__ A 3D shape that has a circle for both bases and a perpendicular line connecting the center of those bases.

Now, it follows from the definition that a circular cylinder’s top and bottom circles will be the same size. So we use to denote the radius of either circle, and we use to denote the height of the cylinder. Without further ado, we present:

__Volume of a Circular Cylinder:__ The volume of a right circular cylinder with a radius of and a height of is

__Surface Area of a Circular Cylinder:__ The surface area of a circular cylinder with a radius of and a height of is

__Practice Problems:__

1. The cylinder below has a radius of and a height of 6. Find .

2. The surface area of a right circular cylinder is and its radius is 2. What is the height of the cylinder?

3. The surface area of a cylinder is twice its volume. Its radius equals its height. What is the volume of the cylinder?

One important kind of 3D shape is the rectangular solid:

which is defined as follows:

__Rectangular Solid:__ A 3D shape with six faces which are all rectangles placed perpendicularly to one another.

The fact that the faces are all perpendicular to each other helps a great deal in calculating the volume, for we simply have:

__Volume of a Rectangular Solid:__ For the following rectangular solid,

its volume is equal to

__Example 1__

Find the area of a rectangular solid whose dimensions are .

__Example 2__

A rectangular solid has two sides that are 3 and 8 units long. It has a volume of 48 units. What is the length of its remaining side?

And, to find the surface area of any rectangular solid, we simply need to add up the area of each of the 6 rectangles which surround it.

__Surface Area of a Rectangular Solid:__ For the following rectangular solid,

its surface area is equal to

How did we get this formula? Well, since each of the faces are perpendicular to each other, we know that there are two rectangles with sides of and units long, one in front and one at the back:

and there are two with sides of units long:

and there are two with sides of units long:

Thus, adding up the area of all of those rectangles, we get: which is the same as what we wrote above.

Now, one common 3D shape is the cube, and the cube is just a special kind of rectangular solid:

__Cube:__ a cube is a rectangular solid whose six sides are all squares.

Our formulas become especially simple for a cube. Let be the length of any side of the cube:

__Volume of a Cube:__ The volume of a cube is .

__Surface Area of a Cube:__ The surface area of a cube is .

These formulas follow directly from just applying our above formulas for the volume/surface area of any rectangular solid.

Finally, it is worth noting how we can use the Pythagorean Theorem to find certain distances on a rectangular solid. Consider the following problem:

__Example 3__

For a rectangular solid with a length of , a width of and a height of , find the length of :

__Practice Problems__

1. Find the length of

2. Find the length of .

3. Find the area of the shaded plane below:

Finally, we turn to the kinds of shapes we see in our daily lives: 3D shapes. Now, by contrast to 2D shapes, 3D shapes appear on relatively few GRE problems, so it may help your scores more to really master the different aspects of 2D shapes.

There are two main 3D shapes that the GRE tests on: the "rectangular solid" (also known as the rectangular prism) and the cylinder (also called the "right circular cylinder").

And there are two main properties of such objects that we will care about:

__Surface Area:__this is the total area of all the surfaces of a 3D shape

So, in a rectangular solid, we add up the area of each of the 6 sides. And, in a cylinder, we add up the area of the circle on the top, the circle on the bottom, and the curved portion in between the two.__Volume:__the space enclosed by a 3D shape

Imagine filling up the 3D shape with water. The amount of space that the water takes up, that’s the volume of the object.

By analogy, you could think of the area of a 2D shape as capturing how much space the lines enclosed. Similarly, volume is a measure of how much space the surfaces of a 3D shape enclose.

In subsequent posts, we will talk about the rectangular solid and cylinder in greater detail and show how to calculate the surface area and volume for each.