Geometry
Tackling Treasure Maps
Treasure Maps are SAT geometry items with a diagram. To do well on these items, you must learn how to fit incomplete pieces of information together to form a conclusion. That’s how SAT geometry works, plain and simple. It’s how 90 percent of all geometry items on the test are designed. Understand the process, and you’ll have SAT geometry licked.
Here’s a four-step method for approaching geometry items with a diagram in them:
Step 1: Combine the information in the diagram with the information in the stem.
Step 2: Determine what information is needed to answer the item.
Step 3: Use basic geometry knowledge to determine values.
Step 4: Give the item what it wants.
Treasure Maps in Slow Motion
Let’s look at each step more closely, using the same example item:
4. If ˚, then what is the area of rectangle ACDE?
(A)
(B) 5
(C) 7
(D) 35
(E) 49
Step 1: Combine the information in the diagram with the information in the stem.
List all the facts in a single place. The easiest way to do this is to add the written information to the existing diagram. For our house example, make a notation that shows angle ABC is 60˚. Every fact in the item is now on the diagram.
Step 2: Determine what information is needed to answer the item.
If you skip this step, the item becomes much harder. The stem asks for the area of the rectangle, but you won’t find it by only looking at the diagram. It’s still well hidden. What you must remember is that the area of a rectangle is found by multiplying its length by its width (A = lw). The length of side CD is provided, so now you need the width. You need to determine either the width of ED or the width of AC.
AC is also the side of the triangle. Alarms should start ringing in your head. Your inner SAT voice should be yelling, “There must be a way to take all the information given about the triangle to find the length of AC.”
Step 3: Use basic geometry knowledge to determine values.
The “basic geometry knowledge” we’re talking about is covered in “Essential Concepts.” In our example, use the following line of reasoning, which assumes some core geometry knowledge:
  • Those little dash marks in AB and BC mean that they are equal in length.
  • Angles that are opposite sides of equal length are also equal, so angle BAC and BCA are the same.
  • Added together, all three interior angles of a triangle always equal 180˚.
  • If angle ABC = 60˚, then angle BAC and BCA must equal 120˚ combined.
  • Since angle BAC and BCA are equal, they must each be 60˚. Every angle in the triangle is 60˚, so every angle is equal, which means that every side is also equal.
  • Therefore, AC must be 5.
You can see how important it is to brush up on the basics. Don’t sweat memorizing every formula, but buckle down and memorize all the facts, because you will need to apply this stuff effortlessly on the day of the test.
Step 4: Give the item what it wants.
Don’t do all the legwork only to goof at the last second. Make sure you come up with the answer the item wants, not a part of the answer or subset of the answer. For example, on our sample item, a student might mistakenly pick choice B since it’s the length of AC. However, the stem is really asking for the area of rectangle ACDE. The area is: A = lw = (7)(5) = 35, choice D. Don’t let the four distractors fool you!
The way you approach these four steps will vary according to the item, of course, but the essential process behind them will remain the same. It may take some time to get comfortable with the four steps, but once you do, you’ll start salivating whenever you see an SAT geometry item. You’ll know what you need to do before you even look at the item.
Guided Practice
Try this one on your own:
5. If ˚, what is the value of ?
(A) 40
(B) 48
(C) 90
(D) 92
(E) 140
Step 1: Combine the information in the diagram with the information in the stem.
Remember, you need all the facts together in one place. This will go a long way toward helping you work through the item.
Step 2: Determine what information is needed to answer the item.
What is the stem asking you for? What other values do you need to determine first? Is there anything “hidden” in the diagram?
Step 3: Use basic geometry knowledge to determine values.
Think back to the essential concepts you learned in the previous section. What values can you figure out?
Step 4: Give the item what it wants.
Make sure you answer what the item is asking you to determine. Don’t get sidetracked by distractors.
Guided Practice Explanation
A bunch of lines get jammed together, but notice that there are no markings saying that any of these lines are either parallel or perpendicular. Let’s use our step method to figure this item out.
Step 1: Combine the information in the diagram with the information in the stem.
The diagram tells you that ˚. From the item, you learn that ˚. Add that fact onto the diagram so that all information is in the same place. Now let’s see what we can do.
Step 2: Determine what information is needed to answer the item.
At first glance it appears that we don’t have enough information to answer this item. Uh-oh, what do we do? Take a closer look at the diagram. Do you notice anything hidden in the jumble of lines? You should see a hidden triangle marked by points A, G, and H. To find the value of one angle in a triangle, you need to know the value of the other two angles. So, to figure out the value of , we have to first know the values of and .
Step 3: Use basic geometry knowledge to determine values.
Let’s go through our line of reasoning:
  • We know that two supplementary angles always equal 180˚.
  • Since you know ˚, you can determine that ˚.
  • We know that two vertical angles are always equal to each other.
  • Since you know that ˚, must also equal 48˚.
Step 4: Give the item what it wants.
Notice that two of the answer choices are 40 and 48. If you’re in a hurry, you might see these familiar numbers and choose A or B, but these answer choices are actually distractors. See the item all the way through, and you’ll reach the right answer.
You now know two of three interior angles of the triangle HAG. Since the stem asks for the third angle (HGA), all you need to do now is a little math:
There’s your answer, choice D.
No part of the stem mentioned the word triangle, yet your knowledge of triangles led you to the right answer. It’s important that you learn to recognize these hidden treasures wherever they appear. The SAT pulls this trick all the time.
Independent Practice
After you complete the following item, turn the page for the explanation.
7. If the area of circle C is , what is the perimeter of triangle CDE?
(A)
(B)
(C) 21
(D) 42
(E) It cannot be determined from the information given.
Independent Practice Explanation
Step 1: Combine the information in the diagram with the information in the stem.
The stem tells you the area, while the diagram states . Add the area to your diagram.
Step 2: Determine what information is needed to answer the item.
To find the perimeter of the triangle, figure out the length of each side. None of these values are provided, so you must determine the length of sides CD, CE, and DE.
Step 3: Use basic geometry knowledge to determine values.
Start with the area information first, since that will give you the radius.
  • We know that the area of a circle = .
  • If , then .
  • So, 7 = r.
  • Looking at the diagram, two of the legs of triangle CDE are radii: CD and CE.
  • So, CD = 7 and CE = 7.
  • The diagram tells you that .
  • CF is also a radius, so it is equal to 7.
  • That means DE = 7.
Step 4: Give the item what it wants.
You now have all the values necessary to solve this item. The perimeter of a triangle is the value of its three sides added together. All three sides of triangle CDE are length 7—an equilateral triangle, who’d of thunk it? Now for a little addition: 7 + 7 + 7 = 21. The answer is C. The distractors in this item aren’t too tricky. If you were rushing, you may have thought A, 7π, was the right answer. But hopefully the π would have given the distractor away.
This item is typical of the kind of mathematical tap dancing required on the SAT, so get used to it.
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