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Re: Triangles
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23 Sep 2018, 23:37
This asks for a specific number for the speed of car Y (beyond point P) as a percent of the speed of car Z. Notice that, while this is a "what is the value" question, it is asking for a ratio rather than a value for the speed of Y. This means that you may be able to find the relationship between the two speeds by leveraging your assets even if you cannot find a specific speed for each car. Remember also that complex situations like this one are often abstractions of fairly straightforward ideas. The test will reward you for breaking down that abstraction into something more concrete. You are given that the roads that cars Y and Z are parallel until point P. At point P the cars are sidebyside. Then car Y makes a turn of x degrees, speeds up, and continues to keep up with car Z. So even though they are going at different speeds, they are travelling to the right (in the figure) at the same relative speed.
Statement (1) gives you the speed of car Z, 50 miles per hour. This statement does not give any information as to the speed of car Y, or the value of the angle x. It should be clear that this statement is not sufficient on its own, and you can prove it by thinking conceptually.
If the two cars had been travelling side by side on the same road before car Y veered off, you would be able to recognize that they seem to form two sides of a triangle with the angle x between them at point P. If angle x is very small, car Y will not have to go much faster than car Z in order to keep up, since it is travelling in almost the same direction.
However, as x increases, the speed of car Y has to increase as well. Thus, without knowing anything about the angle X, you can't make any deductions about the ratio of their speeds.
Eliminate choices A and D.
Statement (2) tells you that the measure of angle x is 45 degrees. While this statement may not appear sufficient, remember that you don't need to know the speeds of the two cars, only the ratio between them. Because the cars are keeping up with each other, the amount of time that each car has had to travel is equal. That means that the increased percentage of distance that car y has traveled will be the same as the increased percentage or the rate of car Y compared to car Z. In other words: determine the ratio of the distance traveled by car Y to the distance traveled by car Z, and you have enough information to answer this question.
Again imagine that cars Y and Z were instead traveling sidebyside on the same road. As with statement (1), when car Y veers off at point P, it forms a triangle whose three sides are the distance between point P and car Y, the distance between point P and car Z, and the distance between cars Y and Z. Because you are told that cars Y and Z remain even, you should recognize that this means that this is a right triangle. Going back to the information given in statement (2), you are told that x is 45 degrees. That means you have a 454590 triangle with the distance that car Y has traveled as the hypotenuse and the distance car Z has traveled as one of the sides.
The ratio of the sides of a 454590 triangle is x:x:x\sqrt{2} where x is a length of one of the legs. The ratio of the distances traveled for Y to Z is the same as the ratio of the length of the hypotenuse to a side, x\sqrt{2}:x which simplifies to \sqrt{2}:1 Since you have established that the ratio of the distances must be the same as the ratio of the speeds, you know that statement (2) is sufficient. The correct answer is B.
