Geometry: Triangles

A 25-foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on concrete feet from the base of the building. If the top of the ladder slips down feet, then the bottom of the ladder will slide out...

The first step in the Knowsys Method is to read carefully. You should notice a few things:
  1. This is a right triangles problem involving the ladder, the ground, and the wall.
  2. The problem provides two sides of the triangle that is formed before the ladder slips: 25 ft and 7 ft.
  3. The 4 ft is not a measure of the second triangle, but instead a measure of change. It tells how far the ladder slipped. Be careful how you use this number! 
After reading carefully, you should look for the Bottom Line. What is this problem actually asking for? In almost all cases, the Bottom Line is found at the very end of the problem text. This one is asking how far the bottom of the ladder slides out. At the top of your scratch work, abbreviate this so you know what you’re looking for.

            ladder base moves = ?

One more piece of Knowsys advice: Be methodical. At first glance, it looks like the answer to this question is 4; after all, if the top of the ladder slid down four feet wouldn't the other end move the same distance? This is a TRAP! If the question were that easy, it wouldn't be on the SAT. Remember that sometimes SAT questions have counter-intuitive answers, so you should be sure to stick with the math instead of what "feels right."

Next, assess your options. Stop and consider “What could I do?” This means you open up your “mental toolbox” to look for any formulas or strategies that could help you. In this case, the Pythagorean Theorem  will probably spring to mind. You could use the Pythagorean Theorem to find the missing side of the first triangle, or you could memorize the Pythagorean Triplets—right triangles with whole numbers on all three sides—and recognize the 7-24-25 triangle in this problem. After that, you will need to find out the dimensions of the second triangle and subtract to find out how far the other end of the ladder moved. Now that you've decided what you SHOULD do, attack the problem fearlessly!

 First, the 7-24-25 triangle shows that the ladder starts off touching the wall 24 feet above the ground. After it slips down 4 feet, it will touch the wall 20 feet above the ground. The ladder, which is also the hypotenuse, is still 25 feet. The problem has conveniently given you another Pythagorean Triplet, the 15-20-25 triangle; after the ladder slips, the base will be 15 feet away from the wall.

This is when many students are tempted to stop. I finished the triangle! I’m done! Loop back to make sure you answered the correct question. You were looking for how far the ladder moves, so there is one more step. The new distance (15) minus the original distance (7) is 8 feet. That is what the question is looking for. Now check the answer choices:

(A) feet
(B) feet
(C) feet
(D) feet
(E) feet
The answer is E. 

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