SAT Math Multiple Choice Question 59: Answer and Explanation
ΔABC is equilateral and ∠AEF is a right angle. D and F are the midpoints of AB and AC, respectively. What is the value of w ?
- A. 1
- C. 2
- D. 2
Correct Answer: B
B There's a lot going on in this problem! But if we take it piece by piece, we'll crack it. Let's start filling in some information. The first thing the problem tells us is that triangle ABC is equilateral. Mark 60 degree angles on the figure. Next, we see that angle AEF is a right angle. Write that in as well. The problem also conveniently tells us that D and F are the midpoints of AB and AC, respectively. Therefore, AD and AF are 2. Finally, the last piece of information reveals that E is the midpoint of DF; mark DE and EF as equal.
Now, what do we have? Triangle AEF is a right triangle, with a hypotenuse of 2 and a leg of 1. Hmm, perhaps the good ol' Pythagorean theorem can help us. Plug the numbers into the theorem, and you'll see that the answer is (B).
You may have also noticed that triangle ADE is a 30°-60°-90° triangle with hypotenuse 2, which means that DE is 1 and w, opposite the 60°, is the square root of 3. In geometry questions on the SAT, there will often be multiple ways to get to the answer. On the day of the test, use whichever way you are most comfortable with.