The figure above shows a square inscribed within an equilateral triangle where the top of the square is parallel to the bottom of the triangle. If the side x of the square measures 12 inches, then what is the perimeter of the equilateral triangle (in inches)? A. 36 B. 36√3 C. 24√3 + 24 D. 24√3 + 36 E. 72
(D) Triangle questions are common on the GMAT, but they are easy if you can break the question down into simple steps and understand the rules associated with the common triangle types. This question breaks down into a few steps:
1) Since the triangle is equilateral, all its sides are the same length. The perimeter will be 3 times the length of one of its sides.
2) The small triangle on top of the square is also equilateral because it is similar to the larger triangle.
3) The sides of the small equilateral triangle must all be 12.
4) Looking at the small triangle to the left of the square, we know that the lower left angle is 60 degrees because it shares the angle with the larger equilateral triangle. Because another angle is 90 degrees, this triangle must be a 30-60-90 triangle.
5) The lengths of the sides of a 30-60-90 triangle are always in the ratio of 1 : √3 : 2. If we divide the height of the triangle by √3, we will have the length of the short side. Multiplying the value of the short side by 2 will give us the length of the hypotenuse: Hypotenuse = (12 / √3) × 2= 24/√3 = (24√3)/3= 8√3.
6) The length of one side of the large triangle is 8√3 + 12.
7) The perimeter is 3 times this amount: 3 × (8√3 + 12) = 24√3 + 36 inches.
The correct answer is choice (D). -------------
Why is the Hypotenuse 24/√3 divided by 3? Wouldn't the length of the side simply be 24/√3 + 12?
The given explanation is quite simple and the rest of simplest variations would be very similar to the given one (with minor changes).
However, let me try to make it even simpler for you:
1. We, basically deal with three facts: - an equilateral triangle (Its sides are all equal and angles equal 60⁰ each) - a right 90⁰-60⁰-30⁰ triangle (Its side, which is opposite to 30⁰ angle, equals 1/2 of hypotenuse. Its side, which is opposite to 60⁰ angle equals √3/2 of the hypotenuse.) - a square (its sides are equal, angles are 90⁰ each)
2. The solution: - All angles of the equilateral triangle are 60⁰. And all angles of the square are 90⁰.
- Therefore the both triangles at the base of the largest one are 90⁰-60⁰-30⁰. The side, which is opposite to the 60⁰ angle, is √3/2 of the hypotenuse. Therefore the hypotenuse is (2/√3) × 12 = 8√3. The side, which is opposite to the 30⁰, is 1/2 of the hypotenuse. It equals 8√3/2 = 4√3.
- Therefore the bottom side of the equilateral triangle is 4√3 + 12 + 4√3 = 12 + 8√3
- The perimeter of the equilateral triangle is 3 × (12 + 8√3) = 36 + 24√3, since its sides are equal.
That's it. -----------
How does this solution differ from the given one? The only difference is NOT dealing with the smaller equilateral triangle (which is formed by the square and the largest triangle). The hardest moment in this solution is the same: we have to deal with 90-60-30 triangle in this question. ---------
Is it possible to solve this question without knowing the properties of 90⁰-60⁰-30⁰ triangle? Yes, however it takes longer. Therefore it's better for you to memorize the properties of 90⁰-60⁰-30⁰ and 90⁰-45⁰-45⁰ triangles.
ALTERNATIVE SOLUTION: 1. The triangle made by the top side of the square and sides of the largest triangle is similar to the largest one (because one angle is the same and the bases are parallel, so the angles at the bases are equal). Therefore the smaller triangle at the top is equilateral as well. All it's sides are 12 inches.
2. Two right triangles at the base are equal because one side of each equals 12 inches and the angles are equal: 90⁰, 60⁰, 30⁰.
3. Let's denote the side of the right triangle at the base of the largest one by x. Then the base side of the largest triangle is (x + 12 + x).
4. All sides of the largest equilateral triangle are equal. So they all are 12 + 2x. Therefore the hypotenuse of the right triangle is 12 + 2x – 12 = 2x.
5. If we apply the Pythagorean Theorem to the right triangle, we get the following equation: (2x)² = 12² + x² x² = 12²/3 x = 12/√3 = 4√3
6. One side of the largest equilateral triangle is 12 + 2x = 12 + 8√3. The perimeter is (12 + 8√3) × 3 = 36 + 24√3.
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