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.
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:
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?
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).