﻿ Find the Missing Pieces in Geometry - Ten Steps to Scoring Higher on the GRE - Crash Course for the New GRE ﻿

## Crash Course for the New GRE, 4th Edition (2011)

### Step 7. Find the Missing Pieces in Geometry

Geometry on the GRE is really like a series of brainteasers. It is your job to find the missing piece of information. They will always give you four out of five pieces of information, and you will always be able to find the fifth. There are only about a half dozen concepts that show up. Once you learn to recognize them, you should be able to handle any problems you might see. The basic problems are made up of parallel lines, triangles, circles, and quadrilaterals.

First, a few basics:

1. Line—A line (which can be thought of as a perfectly flat angle) is a 180-degree angle.

2. Perpendicular—When two lines are perpendicular to each other, their intersection forms four 90-degree angles.

3. Right angle—Ninety-degree angles are also called right angles. A right angle on the GRE is identified by a little box at the intersection of the angle’s sides:

4. Vertical angles—Vertical angles are the angles across from each other that are formed by the intersection of lines. Vertical angles are equal.

 x = y a = b a + b + x + y = 360°

5. Parallel lines—First, never assume two lines are parallel unless they tell you or you can prove it. When you see the symbol for parallel lines, however, there’s generally only one thing interesting about parallel lines. When you see a pair of parallel lines intersected by a third, two kinds of angles are formed: big ones and small ones. All big angles are equal to all big angles and all small angles are equal to all small angles. Any big + any small = 180 degrees. When you see the symbol for parallel lines, always and automatically identify the big angles and the small angles. They are almost assuredly going to come into play. Otherwise, why make them parallel?

b = 140°

Triangles

You know the basics about triangles. All of the angles add up to 180. The largest side is opposite the largest angle, just as the smallest side is opposite the smallest angle. The formula for the area of a triangle is A = (b)(h). An equilateral triangle is one in which all three sides, and hence, all three angles, are the same. The angles on an equilateral triangle are all 60°. An isosceles triangle is one in which two of the sides are the equal. Naturally, sides opposite equal angles are equal in length and vice versa.

A lesser-known rule is that of the third side. The third side of a triangle is always less than the sum of the other two sides, but greater than the difference.

The most frequent triangles you will see are right triangles. A right triangle has one 90° angle and two smaller ones. If you know the length two sides of a right triangle, you can always find the length of the third because of the Pythagorean theorem, which says that a2 + b2 = c2 where c is the hypotenuse (that means that c represents the longest side, which is the one opposite the 90° angle, which is always the largest angle). The Pythagorean theorem tells us that if you add the squares of the lengths of the two shorter sides of a right triangle, it will add up to the square of the longest side.

It’s good to understand how the Pythagorean theorem works, but you rarely need it because there are three kinds of right triangles that come up all of the time. Once you recognize them, you can save yourself some math. So, if you see a triangle on the GRE, be suspicious. If you see a triangle and it is a right triangle, be extremely suspicious. It is likely to be one of three kinds of common right triangles that you should recognize on sight. They are:

Pythagorean Triple—

The Pythagorean theorem will work on any right triangle, but on a Pythagorean triple, a2b2 and c2 are all integers. If you see a triangle, and it’s a right triangle and you see sides of 3 and 4, you’re done. If you see a long side of 13 and a short side of 5, you’re done. Sensitize yourself to these numbers and recognize them on sight.

Right Isosceles—

If you have a square (all sides are equal, all angles are 90°) and you cut it in half on the diagonal, you end up with the triangle you see above. The two sides of the square remain equal. If each side of the square is x, then the diagonal of the square (the same as the hypotenuse of the right triangle) is x. If you see a triangle and there is a  in the problem somewhere, you know what you’re looking for. This means that you also always know the length of the diagonal of a square if you know one side of the square or the area of the square.

30-60-90—

If you take an equilateral triangle and cut it in half, you create the smaller triangle you see above. The angle that never changed, remains 60°. The angle at the top got cut in half and is now 30°. The angle at the base is 90°. One side of the triangle you cut in half, and one side you left the same. If the small side of the new triangle is x, then the big side (opposite the 90°) angle is 2x. The side in the middle, opposite the 60° angle, is x . This is an especially useful triangle because it means that you always know the area of an equilateral triangle because you always know the height. The height of any equilateral triangle is one half of one side times the square root of three. If you see a right triangle and you see a  hanging around the problem anywhere, look for this triangle.

Circles

Circle Facts

·        All circles contain 360°.

·        The radius of a circle is the distance from the center to the outer edge.

·        The diameter is the longest distance within a circle and passes through the center. The diameter is twice the radius.

·        The formula for area of a circle is A = πr2.

·        The formula for circumference (perimeter) of a circle is πd, or 2πr.

With circles, once you know the radius, you know everything. You should be comfortable going from circumference to radius to area and back. Remember that pi equals approximately 3.14 should you need to estimate something, but as a general rule, leave it as pi.

1. Four-sided figure—Any figure with four sides has 360 degrees. That includes rectangles, squares, and parallelograms (four-sided figures made out of two sets of parallel lines).

2. Parallelogram—A four-sided figure made from two sets of parallel lines. The opposite angles are equal, and the big angle plus the small angle add up to 180 degrees.

x = 120°, y = 60°, z = 120°

3. Rectangle—A four-sided figure where the opposite sides are parallel and all angles are 90 degrees. The area of a rectangle is length times width (A = lw).

perimeter = 4 + 8 + 4 + 8 = 24

area = 8 × 4 = 32

4. Square—A square is a rectangle with four equal sides. The area is the length of any side times itself, which is to say, the length of any side squared (A = s2).

A Few Other Odds and Ends

1. Coordinate geometry—This involves a grid where the horizontal line is the x-axis and the vertical line is the y-axis. The x-coordinate always comes first, and the y-coordinate always comes second.

Point A on the diagram above is (2, 4) because the x-coordinate is 2 over from the origin (0, 0) and the y-coordinate is 4 above the origin. Point B is (–7, 1). Point C is (–5, –5).

2. Inscribed—A figure is inscribed within another figure if points on the edge of the enclosed figure touch the outer figure.

3. Perimeter—The perimeter of a rectangle, square, parallelogram, triangle, or any sided figure is the sum of the lengths of the sides:

perimeter = 26

4. Slope—In coordinate geometry, the equation of a line, or slope, is y = mx + b, where the x and the y are points on the line, b stands for the “y-intercept,” or the point at which the line crosses the y-axis, and m is the actual slope of the line, or the change in y divided by the change in xNote: Sometimes ETS uses an a instead of an m.

5. Surface area—The surface area of a rectangular box is equal to the sum of the areas of all of its sides. In other words, if you had a box whose dimensions were 2 by 3 by 4, there would be two sides that are 2 by 3 (area of 6), two sides that are 3 by 4 (area of 12), and two sides that are 2 by 4 (area of 8). So, the surface area would be 6 + 6 + 12 + 12 + 8 + 8, which is 52.

6. Volume—The volume of a rectangular solid is l × w × h (length times width times height). The volume of a circular cylinder is πr2 (the area of the circle that forms the base) times the height (in other words, πr2h).

The Basic Approach

Step 1—Engage the hand. You will need to work with your shapes and fill in lots of little pieces of information, so it is important to get your shape drawn on your scratch paper. Try to draw the shape to scale, but remember to base your information upon what you’re told, not necessarily what you’re shown on screen (on-screen shapes can be misleading). Drawing your shape is particularly important when you’re not actually given a picture of the shape as part of the problem.

Step 2—Fill in what you know. The problem will give you various pieces of information. It might give you an angle or two, the length of a side, or a relationship between two elements, for example. Park all of that information on your drawing.

Step 3—Make deductions. If a problem gives you two angles in a triangle, you can calculate the third. Always do this as a matter of good test-taking habits. The information you’re given will allow you to make some deductions. Sometimes these deductions alone are enough to lead you to the correct answer.

Step 4—Write relevant formulas. Geometry on the GRE is all about finding the missing piece of information. Writing down the formulas you need will help you to organize the information you have and will also tell you which pieces you are missing. This is a trigger and response relationship, just like other parts of the test. When you are working with triangles and you see the word “area,” automatically write down the formula for the area of a triangle. You will end up needing it sooner or later, and it will give you a place to put some of the information you’re given. This is true for any problem that involves a formula.

Step 5—Drop heights/draw lines. When taking your test, you always want to be doing, not thinking. When you get stuck on a geometry problem, first walk away and do a few other problems to distract your brain. When you come back, you need to find another way to view the problem. This is where Step 5 comes in. Try dropping the height of a triangle or parallelogram, drawing in a few extra radii in a circle, or subdividing a strange shape to make two shapes you recognize. Engage your hand and do something. Staring at the problem isn’t going to make it easier.

Here’s an example:

Question 5 of 20

In the triangle above, if BC = 4, then AB =

6

4

8

4

10

Step 2—Fill in what you know. BC is 4. This is a highly suspicious number. No one, not even ETS, randomly decides to make one side of a triangle include a square root. There must be a reason. What kind of triangle uses a ?

Step 3—Make deductions. We know the third angle. It’s 45 degrees. Now you have a 45-degree angle and a  in the same problem. There must be a right isosceles triangle in here somewhere.

Step 4—Write relevant formulas. This problem doesn’t call for any formulas.

Step 5—Drop heights/draw lines. Anytime you drop the height of a triangle, a right angle is formed by the intersection of the height and the base of the triangle. Look at that. There’s your right isosceles! The angle at the top is 45 degrees too. If the hypotenuse is 4, then each of the smaller sides is 4. Plug them into your drawing. On the other side we have another triangle. This one has a 30° angle, a 90° angle. You don’t even have to calculate the third angle; it must be 60°. This is a 30-60-90 triangle, which means that we know the ratio of the sides. The short side, opposite the 30 degree angle is 4. The long side, opposite the right angle, therefore, must be 8. This is side AB, which is the side we were asked to find. The correct answer is (C). Your scratch paper should look like this:

Geometry problems are often like a small piece of knitting. Once you find and tug on the loose thread, the whole thing begins to unravel. The steps are designed to tease out that loose thread.

Geometry on Quant Comp

Question 8 of 20

 Quantity A Quantity B The area of a square region with perimeter 20 The area of a rectangular region with perimeter 20

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

When solving a quantitative comparison with variables, you need to always Plug In more than once. The equivalent of Plugging In more than once with a geometry question is drawing your shape more than once. Ask yourself, “Is there more than one way to draw this shape?”

You can set this problem up just like a quant comp Plug In. Quantity A does not change because there is only one way to draw a square. If the perimeter is 20, then one side is 5, and the area is 25. Now draw a rectangle for Quantity B and make the area as small as possible. You could have a long skinny rectangle with long sides of 9 and short sides of 1. The perimeter is 20, but the area is 9. Cross off choices (B) and (C). Now redraw your shape. How big can you make that area? The biggest you could make it is to make a square with sides of five (a square is a rectangle). The perimeter is 20, and the area is 25. Cross off choice (A). Your answer is (D).

Your scratch paper should look something like this:

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