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:
|y – z| < 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.
|y – z| < 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.
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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