#$&* course Mth 151 3/3 3 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: The original triangle had the three points A, B and C. When you extended the two sides you marked the new endpoints, then you marked the point in the middle of the third side. So you've got 6 points marked. The construction of these numbers is shown in the figure below. We begin with a single dot: We label this point A and construct a triangle containing this point as a vertex. We place similar dots at the vertices of this triangle. We now 'scale up' the triangle by doubling the lengths of its sides: We divide this triangle into triangles of the original size, and place dots at each of these vertices. The first figure has a single 'dot', the second has 3 'dots', and the third has 6 'dots'. Note the similarity with the figures below. The first depicts the pattern illustrated in this question. The second illustrates the pattern extended one steps: The third depicts the pattern as it would appear if extended 12 steps beyond the original triangle: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q002. Extend the two sides that meet at A by distances equal to the original lengths AC and AB and mark the endpoints of the newly extended segments. Each of the newly extended sides will have 4 marked points. Now connect the new endpoints to form a new right triangle. Mark points along the new side at the same intervals that occur on the other two sides. How many marked points are on your new triangle, and how many in the whole figure? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: There are 3 points in the right angle, since each point is supposed to represent the end of the original length and each side of the right angle is to be no more than that. But this means that there are only 2 new points in the figure as a whole, making the total number at this point to be 8. After reviewing the answer, my mistake was that the statement “a new right triangle” made me believe I was to make a triangle with a right angle off of point A, instead of enlarging the original shape even further. After correcting this I was able to come to the right conclusion. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: You added the two new endpoints when you extended the sides. You then should have marked two new points on the new third side, so that each side contains 4 points including its endpoints. Your figure will now contain 10 marked points. The construction is shown below. First we extend the two sides by a length equal to that of the original triangle: Next we join the 'free' endpoints of those new sides to form a triangle. Now we place points along the new side and join them to complete the 'small' triangles within our new figure: We have added four new dots. The figure below depicts only the 'dots', without the triangles: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): After reviewing the answer, my mistake was that the statement “a new right triangle” made me believe I was to make a triangle with a right angle off of point A, instead of enlarging the original shape even further. After correcting this I was able to come to the right conclusion. ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q003. Continue the process for another step-extend each side by a distance equal to the original point-to-point distance. How many points do you have in the new triangle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: With points being added at the same intervals we are to add 5 more to the new side, bringing the total up to 15 points. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: You will add an endpoint to each newly extended side, so each of the new sides will contain 5 points. You will then have to add 3 equally spaced points to the new side, giving you a total of 13 points on the new triangle. In addition there are two marked points inside the triangle, for a total of 15 points. Click on 'Next Picture' to see the construction. The line segments along two sides of the triangle have again been extended and points marked at the ends of these segments. The new endpoints have been connected to form the third side of a larger triangle, and equally spaced points have been constructed along that side. `routine triangle4 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q004. Continue the process for one more step. How many points do you have in the new triangle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: As we continue this pattern at the regular pace, the number of points added to a side goes up one. Since 5 points were added last time, 6 more will be added if we continue at this rate, bringing the total to 21 points. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: You will add an endpoint to each newly extended side, so each of the new sides will contain 6 points. You will then have to add 4 equally spaced points to the new side, giving you a total of 15 points on the new triangle. There are also 5 marked points inside the triangle for a total of 21 marked points. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q005. The sequence of marked points is 3, 6, 10, 15, 21. What do expect will be the next number in this sequence? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The numbers regularly go up by the sequence 3,4,5,6… so we can assume the next number will be 21+7 which equals 28. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The differences between these numbers are 3, 4, 5, 6. The next difference, according to this pattern, should be 7, which would make the next number 28. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q006. How can you tell, in terms of the process you used to construct these triangles, that the next number should be 7 greater? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: As I said before, the number of points increases by one for each original length it duplicates. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: When you extend the triangle again, you will add two new endpoints and each side will now have 7 points. The 7 points on the new triangle will be all of the new points. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q007. How do you know this sequence will continue in this manner? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Each point represents the length of the original side. Continuing in the same manner at the same intervals at the same pace will only continue the pattern. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Each time you extend the triangle, each side increases by 1. All the new marked points are on the new side, so the total number of marked points will increase by 1 more than with the previous extension. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating: The picture below depicts this sequence extended to the 24th number. You should understand why the number of beads in this picture is 1 + 2 + 3 + 4 + 5 + ... + 21 + 22 + 23 + 24. ********************************************* Question: `q008. How many BB's would there be if the top half of the triangle in the above picture was removed? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If half are removed that makes the remaining points to be half of 24, or 12. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The last two questions are a little more challenging than most q_a_ questions. See if you can get them. ********************************************* Question: `q009. A triangle similar to the above triangle, but containing about twice as many rows, would contain 725 BB's. If another row is added there will be 776 BB's. How many rows are there in each of the triangles? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If there are already 24 points, which would be 24 rows, and we were to double that then there would be 48 rows. Then adding one more would make it 49. Riiiiiight? confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: `q010. The formula T_n = 1/2 * n * (n+1) can be used to find the number of BB's in a triangle with n rows (this number is called the nth triangular number). So for example the fourth triangular number is T_4 = 1/2 * 4 * (4 + 1) = 10. The sequence 1, 3, 6, 10, 15, 21, ... of triangular numbers could be written in symbols as T_1, T_2, T_3, T_4, T_5, T_6, ... . Use the formula to find the 14th and the 15th triangular numbers, and calculate the difference between them. Why did you get the difference you did? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: T_14= ˝*14*(14+1)= T_14= 7*(15)=105 T_15= ˝*15*(15+1)= T_15= 7.5*(16)= 120 120-105=15 The difference between the two is 15 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-critique Rating: