You are likely to see a couple of right triangle related questions in almost every GRE test. Therefore, get the right triangle related concepts right!
All of us are familiar with the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs of the right triangle. Diagrammatically, in the following right triangle,
AB2 = BC2 + AC2; often written as c2 = a2 + b2.
Conventionally, the side opposite to angle C is represented by c, the side opposite to angle A is represented by a, and the side opposite to angle B is represented by b.
However, many of us may not be aware that the converse is also true. This means that if in a triangle the square of one side is equal to the sum of the squares of the other two sides, the triangle hasto be a right triangle. This, in turn, leads us to the concept of the Pythagorean Triples.
Any triangle with the sides in the ratio 3:4:5 will be a right triangle because 32 + 42 = 52.
3, 4, and 5 is a Pythagorean Triple. So are all the multiples of 3, 4, and 5 — 6, 8, and 10; 9, 12, and 15; and so on.
While, there are infinite Pythagorean Triples. However, for the purpose of GRE, the most important ones are
3, 4, and 5
6, 8, and 10
5, 12, and 13
Once in a while you may come across 10, 24, and 26 or some higher multiples of 3, 4, and 5.
However, this doesn’t mean that in a right triangle the sides must always have integer values! Rather, Pythagorean Triples are just integer solutions to the equation c2 = a2 + b2.
Two other types of right triangles are very common on the GRE. These are 45-45-90 triangles and 30-60-90 triangles.
If in a right triangle, the three angles measure 45o, 45o, and 90o, the corresponding (opposite) sides will be in the ratio 1: 1: √2.
These triangles are also referred as isosceles right triangles.
It’s important to note that an isosceles right triangle is half of a square. This leads to other relationships such as
- in a square, the diagonal is √2 times any side of the square and
- the area of a square is half the diagonal’s square (½ d2 – where d is the length of the diagonal)
Another type of triangles that are very common on the GRE are the 30-60-90 triangles. If in a right triangle, the three angles measure 30o, 60o, and 90o, the corresponding (opposite) sides will be in the ratio 1: √3: 2.
It’s important and interesting to note that a 30-60-90 right triangle is half of an equilateral triangle. There are a few corollaries to this.
- The area of an equilateral triangle with side a is given by
- In any right triangle, if one leg is half the hypotenuse, the triangle has got to be a 30-60-90 triangle and that leg will be the one opposite to the 30o angle.
These 45-45-90 and 30-60-90 triangles, including the right triangles corresponding to Pythagorean Triples, are often referred to as special right triangles.
Special Right Triangles and GRE
The GRE is sort of obsessed with Pythagorean triples. Here are a few examples.
Try to solve the questions yourself, and then look at the notes that follow.
Q. No. 1: In a coordinate plane what is the distance between the points A (3, 3) and B (8, −9)?
Note: As soon as we see this question, we may be tempted to use the distance formula. Wait a moment. As you look at the positive difference between the values of the x-coordinates and the positive difference between the values of the y-coordinates, you find that the differences are 5 and 12 respectively. Therefore, the line AB is the hypotenuse of a right triangle with legs measuring 5 and 12, and, therefore, the distance has got to be 13. If you are conscious of GRE’s obsession with the Pythagorean Triples, you are better off!
Here is another one.
Q. No. 2: In the circle above, AC is a diameter, B is a point on the circumference, and AB < BC. If, AC = 20 and the area of the triangle is 96, what is the length of AB?
Note: The angle in a semi-circle is a right angle. Therefore, the triangle ABC is a right triangle right angled at B. Many of us may be tempted to write the relationships = 96 and AB2 + BC2 = 202 and proceed to solve the equations, but wait a moment. If AC (the hypotenuse) = 20, which Pythagorean Triple it corresponds to? 3, 4, and 5 multiplied by 4. That is 12, 16, and 20. Why don’t we try it out before we start solving the set of equations? × 12 × 16 = 96. Voila! You have got it! AB = 12. GRE’s obsession with Pythagorean Triples works to our advantage.
Here is another weird way or rather surreptitious way in which GRE may test your alertness with regard to Pythagorean Triples.
Figure not drawn to scale
Q. No. 3: The figure above has two circles with the shaded region entrapped between them. If the radius of the larger circle is 13 and the area of the shaded region is 144π. What is the radius of the smaller circle?
Note: If you can figure out that 144 is 122, you really do not have to do anything other than recalling the Pythagorean Triple 5, 12, and 13.
52π (Area of the smaller circular region) + 122π (Area of the shaded region) = 132π (Area of the bigger circular region). The answer is 5.
Once you know that GRE is obsessed with Pythagorean Triples, you will be able to use it to your advantage. Knowledge of such nuances helps you ace the GRE; in our classes at Manya – The Princeton Review, we impart just that knowledge.
More in Part II of this Article.
By the way, what do you see in the numbers below?