GMAT Exam: A Simple Guide To Solving Word Problems - Right Triangles

3-4-5 Right Triangles
You will likely see right triangles on the GMAT that have sides of lengths 3, 4, and 5 or of lengths that are multiples of 3, 4, and 5.

The reason is that 3, 4, and 5 and any multiple of these numbers (such as 6, 8, and 10 or 30, 40, and 50) satisfy the Pythagorean theorem.

That is, 32 + 42 = 52, 62 + 82 = 102, and 302 + 402 = 502.


Example: A wall is 12 feet long and 9 feet high. What is the diagonal length of the wall?
The diagonal is the hypotenuse of a right triangle that has legs of 12 feet and 9 feet. The numbers 12 and 9 are multiples of 4 and 3 by a factor of 3, respectively. Therefore, the hypotenuse is a multiple of 5 by a factor of 3. Thus, the diagonal length of the garden is (3)(5 feet) = 15 feet.

 

30°-60°-90° Right Triangles
In a right triangle that has acute angles of 30° and 60°, the lengths of the triangle’s three sides are in the ratio or, equivalently, .


Example: Find BC in the right triangle shown below.

The triangle is a 30°-60°-90° right triangle. is opposite the 30° angle.

Therefore, .

45°-45°-90° Right Triangles
In an isosceles right triangle that has congruent acute angles of 45°, the lengths of the triangle’s three sides are in the ratio or, equivalently, .


Example: Find the length of the hypotenuse in the right triangle shown below.

The triangle is a 45°-45°-90° right triangle. The three sides are in the ratio .

Therefore, the length of the hypotenuse is .