Triangles dominate GRE geometry. Make sure you learn the area formula for a triangle (½ base × height) plus facts like "the sum of a triangle's interior angles is 180°." But you probably already know about that stuff. Here are some GRE triangle facts you may not know about.
The height of a triangle can be measured on the outside.
To find a triangle's height, you measure the length of a line segment that starts at one corner—a vertex—and meets the opposite side—the base—at a 90º angle.
You're probably used to drawing a height line on the inside of a triangle that runs from the top corner to the bottom side. But what if that line doesn't meet the bottom of the triangle at a right angle? You can draw the height line on the outside so that it's perpendicular to the the plane of the bottom side.
The base of a triangle can be any side.
Or you can pick a side other than the bottom to be the base. Although "bottom" and "base" are synonyms in everyday speech, in geometry jargon a triangle's "base" is whichever side you want. So, if you prefer, pick a side that places the height line inside the triangle.
The height of an equilateral triangle is one-half the side length times the square root of three (½ s√3).
For a triangle with equal sides, finding the height is easy. You just need to know the side length and these two basic triangle facts.
- An equilateral triangle can be split into two equal 30-60-90 triangles.
- A 30-60-90 triangle has sides in a ratio of a : a√3 : 2a, where a is the shortest leg and 2a is the hypotenuse.
In an equilateral triangle, the side length s equals 2a, and the height h equals a√3, so the height works out to ½ s√3.
- h = a√3
- 2a = s
- a = ½ s
- h = ½ s√3
You don't need to memorize this proof, of course. It just shows you why the one part you do need to remember—h = ½ s√3—will give you the height of an equilateral triangle once you know the side length.
The diagonal of a square is the side length times the square root of two (s√2).
Speaking of splitting shapes into right triangles, finding the diagonal of a square comes down to knowing its side length plus this pair of triangle facts.
- A square can be split into two equal 45-45-90 triangles.
- A 45-45-90 triangle has sides in a ratio of s : s : s√2, where s is the leg and s√2 is the hypotenuse.
The sides of a square are the same as the legs of the right triangles it contains, so a square's diagonal d must be the hypotenuse that those two triangles share. So make sure you memorize that d = s√2 for a square.
Right triangles on the GRE often have sides in a ratio of 3:4:5 or 5:12:13.
Side-length ratios for right triangles are a recurring theme in this post—and for good reason. GRE Quant favors not just triangles but often right triangles in particular. 30-60-90 and 45-45-90 triangles, both named for their angles, are two common types.
Two more are 3:4:5 and 5:12:13 right triangles, both named for the ratio of their side lengths. Each of these ratios forms what geometry junkies call a "Pythagorean Triple" or whole-number solution to the Pythagorean Theorem: a2 + b2 = c2.
Memorize 3:4:5 and 5:12:13 and, in many cases, you won't need the Pythagorean Theorem. For instance, if you need to find c when a and b are 6 and 8, don't use the theorem: recognize the ratio.
- 6 : 8 : c = 2(3) : 2(4) : 2(5)
- 6 : 8 : c = 2(3) : 2(4) : 2(5)
- 6 : 8 : c = 10
The legs 6 and 8 are twice 3 and 4, so the hypotenuse must be twice 5 and, thus, 10.
Triangles hide in other figures in GRE Quant.
You'll see shapes besides triangles on the exam, such as circles, squares, and more. Yet even then a triangle may be lurking. For instance, if you get a question with a square inscribed in a circle, get ready to work with a 45-45-90 triangle. In fact, whenever you encounter right angles (in a square or elsewhere) in a problem, consider solutions that involve right triangles.
Did you find this helpful? You can find more triangle tips in the PowerScore GRE Quantitative Reasoning Bible!Photo: "ABC Museo Madrid" courtesy of Paul Thompson.