# Triangles

Yesterday's Question of the Day about Red Cloud piqued my interest, so I decided to look him up for today's Link of the Day. Red Cloud was an amazingly successful war leader of the Lakota Indians, assaulting several United States Army forts along the Bozeman Trail in the 1860's. By the end of the decade, the US agreed not only to abandon its forts in Lakota territory, but also to guarantee Lakota control over a vast land area, including the western half of modern South Dakota and parts of Montana and Wyoming. Unfortunately, Red Cloud's victories did not last, and eventually the white settlers reclaimed and broke apart the Lakota holdings. Red Cloud's tireless efforts to protect his people and his culture would make an outstanding Excellent Example for your essay.

Let's take this a step further: What kind of essay prompt could you answer with the story of Red Cloud? Please respond in the comments!

## 7/26 > Triangles

Whenever you approach a math problem, remember to follow the Knowsys method. Rather than charging in, take a moment to read the problem carefully and identify the bottom line. Consider the best way to approach the problem--what could I do? What should I do? Then attack the problem and, finally, loop back to the top and make sure you answered what the question was actually asking. The last and easiest step is to match your answer to the provided answer choices.

In triangle ABC, the length of side $\overline{BC}$ is 2 and the length of side $\overline{AC}$ is 12. Which of the following could be the length of side $\overline{AB}$?

First, note the bottom line at the top of your scratch work.

$\overline{AB}$ = ?

Next, consider your options. What does the problem tell you? What strategies, formulas, or theorems do you know that could help you solve it? In this case, the problem tells you that you are dealing with a triangle and supplies two side lengths. With so little information, you really only have one tool that can help you: the Triangle Side Lengths Inequality.

The Side Lengths Inequality states that any side of a triangle must be less than the sum and greater than the difference of the other two sides. When you think through it, this actually becomes fairly obvious. If one side were longer than the other two sides put together, the shape could no longer be a triangle. It would fold flat into a line. If one side were too short, it would not be able to "reach" the other sides and the triangle would just be three line segments rather than a closed shape. The Side Lengths Inequality is usually expressed this way:

$\left | y-x \right |

For simplicity's sake, you can rename the sides of the triangle in the problem x, y, and z rather than shuffling As, Bs, and Cs around. (Be sure to note this in your bottom line!) Now that you've chosen the most efficient way to solve the problem, attack it ruthlessly!

First, take the side lengths you are given and plug them into y and z.

$\left | 2-12 \right |

Next, perform some simple arithmetic to solve for x.

$\left | -10 \right |

$10

Now you've narrowed down the range of possible values of x. Loop back to double-check the bottom line. If you remembered to update it earlier, it should look something like this:

x = $\overline{AB}$ = ?

Since you've found the possible values of x, you've also found the possible lengths of side $\overline{AB}$. The last step is to find an answer choice that matches what you found.

(A) 6

(B) 8

(C) 10

(D) 12

(E) 14

Note that the inequality uses "less than" signs, not "less than or equal to" signs. That means that side $\overline{AB}$ cannot equal 10 or 14; it must be 12. The answer is D.

On sat.collegeboard.org, 45% of responses were correct.

For more help with math, visit www.myknowsys.com!