Grades 9-10 Video Solutions 2014-Levels 9&10 Video Solutions 2014 problem12

Levels 9&10 Video Solutions 2014 problem12

Video Transcription

Question number 12. Paul put some rectangular pictures on the wall. For each picture, he put one nail into the wall right here, two and a half meters above the floor, and attached  two meters of string at the two upper corners of the picture. Which of the following pictures  is closest to the floor? And here the answers are given in the following format. The first number is the width of the picture, and the second number is the height. So we have the  following observations to make. First of all, the triangle that we form here by stretching string on the nail and attaching it to the upper right-hand corners is isosceles, and  that is one meter here and one meter here. This string is exactly equally spaced. So in our possible answers, the hypotenuse in red is exactly 100, and that stands for one  meter or 100 centimeters in each case. Then we have two legs of this triangle here in each case, and what I'm referring to is half the width. Okay, so that is labeled here w  over two, and then we have a lowercase h, and the height here given in the problem is capital H. So what we have to do is compute H in each case, add H, capital H, and pick  the largest answer. That will represent the longest distance from the nail to the bottom of the picture, and so such an arrangement must mean our picture is hanging closest to  the floor. So let's write in the values here of w over two. We have 60 divided by two, so 30 over here, then 60, 60, 80, and 80, and now using the Pythagorean theorem, we have that  H here is going to be equal to the square root of 100, in the first case, minus 30 squared,  which comes out to be equal to the square root of 9100, and that is, for our intents and purposes,  about 100, less than 100 in fact. So that's what we will need later. For the other triangles,  we use the following relationship. We have 3 squared plus 4 squared is 5 squared, and then we can  multiply by 20 to obtain 60 squared plus 80 squared is 100 squared, and we have our values  of H here, 80, 80, 60, and 60. So we see that there are a couple of choices here that we can focus on  right away. In C, we have 80 plus 90, or 170, and B, we have 80 plus 50, which is 130, and then in A,  we have the square root of 9100 plus 40, which is going to be less than 100 plus 40. Then, just to  be complete, in D, we have 60 plus 60, 120, and E, we have 60 plus 100, or 160. So the largest here value  is in choice C, and that represents the greatest distance from the nail to the bottom of the  picture, hence also that picture hangs closest to the floor. So we choose the dimensions here in C, and that is our answer.

Video Summary

Paul hung a series of rectangular pictures using the same method, with a nail placed 2.5 meters above the floor and 2 meters of string attached at the upper corners. The task was to determine which picture hangs closest to the floor by calculating the longest distance from the nail to the bottom edge for each picture, forming isosceles triangles. Using the Pythagorean theorem, each picture's dimensions were analyzed. Choice C had the largest combined height calculation, making it the picture hanging closest to the floor. Therefore, option C is the chosen answer.