GMAT Quantitative Review

3.0 Math Review

3.3 Geometry

6. Triangles

There are several special types of triangles with important properties. But one property that all triangles share is that the sum of the lengths of any two of the sides is greater than the length of the third side, as illustrated below.

image

An equilateral triangle has all sides of equal length. All angles of an equilateral triangle have equal measure. An isosceles triangle has at least two sides of the same length. If two sides of a triangle have the same length, then the two angles opposite those sides have the same measure. Conversely, if two angles of a triangle have the same measure, then the sides opposite those angles have the same length. In isosceles triangle PQR below, image since image.

image

A triangle that has a right angle is a right triangle. In a right triangle, the side opposite the right angle is the hypotenuse, and the other two sides are the legs. An important theorem concerning right triangles is the Pythagorean theorem, which states: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

image

In the figure above, image is a right triangle, so image. Here, image and image, so image, since image and image. Any triangle in which the lengths of the sides are in the ratio 3:4:5 is a right triangle. In general, if a, b, and c are the lengths of the sides of a triangle and image, then the triangle is a right triangle.

image

In image triangles, the lengths of the sides are in the ratio image. For example, in image, if image, then image and image. In image triangles, the lengths of the sides are in the ratio image. For example, in image, if image, then image and image.

The altitude of a triangle is the segment drawn from a vertex perpendicular to the side opposite that vertex. Relative to that vertex and altitude, the opposite side is called the base.

The area of a triangle is equal to:

image

image

In imageimage is the altitude to base image and image is the altitude to base image. The area of image is equal to

image.

The area is also equal to image. If image above is isosceles and image, then altitude image bisects the base; that is, image. Similarly, any altitude of an equilateral triangle bisects the side to which it is drawn.

image

In equilateral triangle DEF, if image, then image and image. The area of image is equal to image.