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Triangles

Geometry: Triangles

Read the following SAT test question and then select the correct answer. 

You should start by reading the problem carefully and identifying your bottom line.  Then assess your options and choose the most efficient method to attack the problem.  Finally, loop back to make sure that the answer you found matches the bottom line that you set out to find.

Which of the following CANNOT be the lengths of the sides of a triangle?

Bottom Line: The word “CANNOT” tells you that you are looking for something that is not true.

Assess your Options:  Problems that ask you to find what cannot be true are often impossible to predict; you will have to look at the answer choices to determine whether they work.  Go ahead and look at the answers.

(A) 1, 1, 1
(B) 1, 2, 4
(C) 1, 75, 75
(D) 2, 3, 4
(E) 5, 6, 8

You could try to use logic and your experience with triangles to eliminate some choices.  For example, answer choice A is not the answer because you know there is such a thing as an equilateral triangle and the sides 1, 1, and 1 would create that kind of triangle.  However, after that point you would probably just be guessing.  If you know the Triangle Inequality Theorem, you can systematically check each answer choice.

Attack the problem:  When you are dealing with three sides of a triangle and you do not know that the triangle is a right triangle, you should always think of the Triangle Inequality Theorem.  This theorem states that for any triangle, side x is less than the sum and greater than the difference of the other two sides.  In other words, each side of the triangle must be less than the other two sides added together and greater than the difference of the other two sides.  If you have a triangle with sides x, y and z, you would write the theorem this way:
|yz| < x < y + z. For the subtraction part you can use absolute value or just always do the bigger side minus the smaller side; the result will be the same. The easiest way to think about the theorem is this: for any triangle, other sides subtracted < one side < other sides added.

Start by checking (E).  It has sides 5, 6, and 8.  Plug these sides into your formula by using the first side, 5, as your x.  Make sure that 5 is greater than the difference of the other two sides, but smaller than the other two sides added together.

|y – z| < x < y + z
8 - 6 < 5 < 6 + 8 
2 < 5 < 14
This is true!  (E) works as a triangle.

Now check (D). It has sides 2, 3, and 4.  Plug those numbers into the Inequality Theorem.
 |y – z| < x < y + z
4 – 3 < 2 < 3 + 4
1 < 2 < 7
This is true!  (D) works as a triangle.

Now check (C).  It has sides 1, 75, and 75.  Plug those numbers into the Inequality Theorem.
|y – z| < x < y + z
75 – 75 < 1 < 75 + 75
0 < 1  < 150
This is true!  (C) works as a triangle.

Now check (B).  It has sides 1, 2, and 4.  Plug those numbers into the Inequality Theorem.
|y – z| < x < y + z
4 – 2 < 1 < 2 + 4
2 < 1 < 6.
Is 2 less than 1?  No!  This is false.  You cannot have a triangle with these three side lengths.

Loop Back:  Your bottom line was to find an answer choice that cannot be a triangle, so you are finished!

The correct answer is (B).

On sat.collegeboard.org, 49% of the responses were correct.

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