Triangle Properties to Know for the CAT

Triangle properties are key in understanding geometric problems on the Common Admissions Test (CAT). This is because more complicated diagrams can often be reduced to triangles. After reading this post, you’ll get what you need to know about triangle properties in order to succeed on the CAT!

Penrose Triangle sculpture - public domain image

I wonder if this triangle will show up in the CAT… (By Samo Kupper [Public domain], via Wikimedia Commons)

Basic Triangle Properties

Let’s review the basics. It may help to refer to the following diagram.

Triangle

  • The sum of interior angles is always 180 degrees. That is, x + y + z = 180.
  • Triangle Inequality. The sum of any two sides is always greater than the third side, and the difference is always less than the third side. So we have a + b > c, and |ab| < c.
  • The side opposite the largest angle is the largest side, and the side opposite the smallest angle is the smallest side. That is, if a < b < c, then x < y < z.
  • A perpendicular line drawn from a vertex to the base is called an altitude and measures the height of the triangle. Once you know the height, you can find area using Area = (1/2) Base × Height, or more concisely, A = bh/2.
  • The perimeter of the triangle is the sum of its sides, or P = a + b + c.
  • Two triangles are similar if their corresponding angles are the same. If Triangle ABC is similar to Triangle DEF, then the ratios AB : DE, AC : DF, and BC : EF are all the same.
Similar Triangles

Similar triangles

Special Triangles

A number of properties apply only to specific kinds of triangles.

  • An equilateral triangle has all three sides congruent. As a consequence, all angle are congruent, and they each measure 60 degrees. The area of an equilateral is (√3)/4 × s2, where s is the side length.
  • An isosceles triangle has two sides congruent. The angles opposite the equal sides must also be equal.
  • A scalene triangle has no congruent sides.

Right Triangles

A right triangle must have a right angle (90 degrees). The side opposite the right angle is the hypotenuse, and the other two sides are called legs.

Right triangle properties

Right triangle

Pythagorean Theorem. In a right triangle, a2 + b2 = c2, where a and b are the leg lengths and c is the length of the hypotenuse.

There are two special right triangles that are useful to remember.

The 30-60-90 triangle is essentially half of an equilateral triangle. It’s side lengths are in ratio 1 : 2 : √ 3.

The 45-45-90 triangle is the only triangle that is both right and isosceles. It’s side lengths are in ratio 1 : 1 : √ 2.

Special Triangles

Left, 45-45-90 triangle; right, 30-60-90 triangle. These are your friends!

For more about right triangles, check out this article from Magoosh: Conquering Right Triangles and the Pythagorean Theorem.

Conclusions

You’ll find a good number of geometry problems on the CAT, so it’s important to understand the fundamentals, including triangle properties. With enough study, you’ll ace all of the geometry problems on the CAT!

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