There will always be some questions related to the right-angled triangle in every GRE test. Hence, you must have the right triangle related concepts on tips!
The Pythagorean Theorem is a basic concept which states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the length of the other two sides of the triangle. Diagrammatically, in the following right triangle,
AB2 = BC2 + AC2; also 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.
Another concept that emerges from this theorem is the concept of Pythagorean Triples. This essentially means that a triangle will be a right-angled triangle if the sum of one side is equal to the sum of squares of the other two sides.
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Any triangle with the sides in the ratio 3:4:5 will be a right triangle because of 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.
Even though there are infinite Pythagorean Triples, one has only to remember the significant ones, which 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! Instead, Pythagorean Triples are just integer solutions to the equation c2 = a2 + b2.
Two other types of right triangles are prevalent on the GRE. These are 45-45-90 triangles and 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.
An important thing to note that a 30-60-90 right triangle is half of an equilateral triangle. There are a few corollaries to this.
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 to as isosceles right triangles.
The fact that an isosceles right triangle is half of a square leads to other relationships such as:
The above two triangles along with the Pythagorean Triples are known as special right triangles.
It is seen that the GRE includes several Pythagorean Triples related questions.. Here are a few examples.
Try to solve the questions yourself, and then look at the notes that follow.
Here is another one.
No. 1: 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.
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Here is another weird way or rather a surreptitious way in which GRE may test your alertness about Pythagorean Triples.
Figure not drawn to scale
No. 2: 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 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 more prominent circular region). The answer is 5.
No. 3: 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!
After you get a knack of the types of questions GRE includes concerning Pythagorean Triples, you can easily utilize this as your advantage. Knowledge of such nuances helps you ace the GRE preparation; 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?
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