February 25, 2014

Fun with Triangles

The Question:  Can any 3 side lengths form a triangle?

Well, that is the age-old question when you are looking at triangle problems. Triangles regularly appear on the SAT, and with knowledge of some basic properties, you can easily rock these questions. A triangle is a three-sided polygon where the sum of the interior angles is always 180 degrees.

The formula for determining the area of a triangle is: A=(1/2) bh, where b=base and h=height. Although this formula is well-known, students tend to forget that the base and the height of the triangle must always form a 90-degree angle.

 Triangle 1

Keep in mind, though, that a triangle’s height can be located either inside or outside it.

The Triangle Inequality Theorem states that for any side of a triangle, its value must be between the sum and the difference of the other two sides.  You can find a cool demonstration of this theorem here. Let’s consider the following example:  If we have a triangle with two sides, 3 and 7, can 6 be the value of the third side?

We know the sum of 3 + 7 is 10, and the difference of 7- 3 is 4, so the third side must be between 4 and 10. Since 6 is between those two numbers, then 6 can be the value of the third side.  Neat, right?

In summary, if a, b and c are the sides of a triangle, then:

Triangle 2

a + b > c

b + c > a

a + c > b

Using the Triangle Inequality Theorem, solve the following problem:

In triangle XYZ, the length of side ZY is 11, and the length of side XZ is 17. What is the greatest possible integer length of side XY?

  1. 9
  2. 17
  3. 26
  4. 28
  5. 27

Solution:

Applying the Triangle Inequality Theorem, we have the following:

The sum of the two given sides:

Since the sum of the two given sides is 28, the greatest possible length of the third side would be (E).

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