Area of the smaller square - July 22-26, 1996
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One altitude of an equilateral triangle is a side of one square, and one side of the same equilateral triangle is a side of a second square. The area of the larger of these squares is 56. What is the area of the smaller of the two squares?
It seemed as if all of South Fremantle Senior High School in Australia put up Web pages illustrating solutions to the problem this week, and it was hard to choose among them! Following are three of the earliest and best submissions, featuring both discussions and drawings. Indiana, Hawaii, and Dartmouth '92 are also represented.Chid Gilovitz Grade: 8 School: South Fremantle Senior High Hi! I wrote a web page on the July 22-26 POW. You will find it at gilovitc7.22.96.html. Seeya! Chid
Heather-Jayne Kirkby Grade: Grade 8 School: South Fremantle Senior High School This is the first time I have entered any of your problems although I have figured out and completed many. I find most of them very challenging and interesting. I hope to submit more of my answers to your problems in the future. You will find my entry at kirkby7.22.96.html. Thank you for supplying me with the problem. See you next time! Heather-Jayne
Paul Malone Grade: 9 School: South Fremantle Senior High School I have never before sent an entry in to these problems so I am submitting this solution. In the past I have received a printed copy of many problems but I never completed a solution within the time limit. I have them all presented in a file, some completed and some incomplete, so I can look back when I am stuck. This week I finished because my mathematics teacher gave us time in class. My solution is not terribly accurate but I think that it will suffice. You will find it at malone7.22.96.html. I hope you find the answer and my working out all right.
Bryce Johnson Grade: between 10 and 11 School: Center Grove High School, Greenwood, Indiana Because the side of an equilateral triangle is longer than the altitude, the square adjacent to the side is the larger square. Therefore the side of the equilateral triangle is radical 56, or 2 radical 14. The altitude creates a 30-60-90 right triangle with the altitude as the longer leg. Dividing by 2 and multiplying by radical 3 gives the length of the altitude as the square root of 42, which means the area of the smaller square is 42.
Jason Yeung Grade: 9 / 10 School: Iolani School The area of the smaller of these squares is 42. First of all, in an equilateral triangle, if the side is 2x, then the altitude of the triangle is x* sqrt 3. We can prove that because an equilateral triangle is composed of two 30-60-90 right triangles. Since 2x is larger than x*sqrt 3, the area of the square with the side of the triangle as its side is bigger than the one with the altitude. If the area of that square is 56, then its side will be sqrt of 56, or 2* sqrt of 14, which is also the length of the side of the triangle. If 2x (the side) is 2* sqrt of 14, then x is sqrt of 14, and x*sqrt of 3 is sqrt of 42, which is the altitude and the side of the smaller square. Since the side of the smaller square is sqrt 42, the area of the square is sqrt of 42 * sqrt of 42, which is 42, the answer to the question.
Brian Gordon Dartmouth '92 The altitude of an equilateral triangle divides the triangle into 30-60-90 triangles. Therefore, it's easy to see that the ratio between the sides of the squares is sqrt(3) : 2. Squares are all similar, and the ratio of areas of similar figures is equal to the square of the ratio of corresponding sides. Squaring sqrt(3) : 2 gives an area ratio of 3:4 Since the big square has ratio 56, we have 3:4 = x:56 So the area of the little square is 42. --bri
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