## Geometry: Triangles

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

Always read
the question carefully so that you don’t misapply any information. Identify the bottom line and assess your
options for solving the problem. Choose
the most efficient method to attack the problem. When you think you have the answer, loop back
to make sure that it matches the bottom line.

*If triangle ABC above is congruent to triangle DEF (not shown), which of the following must be the length of one side of triangle DEF?*

**Bottom Line**: side of

*DEF*= ?

**Assess your Options**: Many students go straight to the Pythagorean Theorem whenever they see a right triangle. This formula,

*a*² +

*b*² =

*c*², will not help you in this case because you do not know

*a*or

*b*. Instead, use your knowledge of special triangles to solve this problem.

**Attack the problem**: As soon as you see that this is a 30° – 60° – 90° triangle, you should think about the sides that relate to this special triangle. Those sides, which you should have memorized, are

*x*-

*x*√3 – 2

*x*. Remember that the longest side has to be across from the biggest angle, the 90° angle. That is your 2

*x*. This triangle has a 12 in that position. Solve for

*x*.

2

*x*= 12*x*= 6

Now you know that the side
across from the 30° angle,

*AB*, must be 6. Label it. Look at the side across from the 60° angle.*AC*must be*x*√3. You know that*x*= 6, so this side must be 6√3. Label it. You now know all the sides of triangle*ABC*:*x*-

*x*√3 – 2

*x*

6 -6√3 – 12

Your bottom line is a side
on triangle

*DEF*, not on triangle*ABC*. However, the problem tells you that*ABC*is congruent to*DEF*. Congruent triangles have the same shape and size; they are basically the same triangle with different labels. That means that the side lengths from triangle*DEF*will match the lengths you already found for*ABC*.**Loop back**: You took into account all of the information that you were given and solved for your bottom line. Look down at your answer choices. One of the three side lengths you found will be there.

(A) 18

(B) 24

(C) 3√6

(D) 6√3

(E) It cannot be determined from the
information given.

The correct answer is (D).

On sat.collegeboard.org,
38% of the responses were correct.

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