A Math Forum Project

Geometry Forum - Problem of the Week Solutions - The Rectangles of a Tennis Court, Oct. 24- 28, 1994 Annie says: A lot of answers submitted for this one, but only three of them really hit the nail on the head, without doing a lot of extra stuff: Matthew Phillips, Susan Garges, and Kim Biederman said that all you need to check is whether the diagonals are the same length. Everyone else came up with viable answers about constructing right triangles using 3-4-5 triangles and whatnot, and others mentioned the diagonals fact as part of a longer answer, and they're right, but it's then tough to be sure you're extending your sides "straight". I liked this problem because it reminded me of the summer I spent building a house - the head carpenter (this was a crew of three), while "less educated" than I was (I had finished three years of college, and of course figured I knew everything), taught me a lot of practical geometry. "Gotta check the foundation for square," he said the first day, and whipped out his Really Long Tape Measure. We checked the diagonals, and since they were within 3/4" over the whole building, we decided we didn't need to do major changes before laying down the sill - we'd measure it again then and correct the problem. Pretty cool! And simple...and very useful. I guess what I'm trying to say is that sometimes there is more than one answer to a problem, and while at times it doesn't matter if they're different, it's often useful to be able to find the simplest solution when you're actually going to use the method to check something. Matthew T. Phillips _________ |\ /| The diagonals of a rectangle must be the same, | \ / <-- Diagonal a quality often used in construction. All | \ / | Boutros and Bernice need to do is measure out |---X---|<-- Midcourt sides of the rectangle and then run string | / \ | between the corners as shown to the left. This | / \ | will ensure the correct dimensions and angles. |/ \| --------- The principal application for this, as I said above, is construction. It is difficult to use a T-square with long distances, and most carpenters don't keep surveyor's transits with them, and therefore use this method. Matt Sowa The dimensions of the tennis court is a given fact. The court is 78 feet long and 36 feet wide. We will try to figure out the length of the diagonal line through the center of the court. We will call that line "C" along with "A" for the length and "B" for the width. I will be using the Pythagorean theorem equation A2 + B2 = C2. A2 is equal to 6084. B2 is equal to 1296. C2 is equal to 6084 + 1296; therefore, C is equal to 86 feet. Now that we know the length of the 3 lines you would cut pieces of string to equal them. Then you would lay them out and when they come together they will form right angles. Heather Dankmyer To lay out the triangle I would take the string and cut a 4ft, 3ft, and 5ft piece of string to find out the 90 degree angle at the corners. I would measure out the first angle by judging a 90 degree angle with the 4ft and 5ft of string. Then take the 5ft of string and make a triangle with the third piece. If the 5ft piece of string is overlapping the 4ft or 3ft piece of string, then the angle is less than 90 degrees and you must start over. If the 5 ft piece of string is not reaching the 3 ft or 4ft piece of string the angle is greater than 90 degrees and you must judge the angle again. To reach the measurements of 3, 4 and 5 ft, I used the formula: a2 + b2 = c2. To double check that all of my angles are 90 degrees I would take string and lead it from one corner to an opposite one and do that with the other two corners and measure both strings and if they are even, then all the angles are even, but if the strings don't measure up equally then one or more of your angles are off. Eric Wahl If length of rectangle is L and width is W: cut a piece of string and make a closed loop with perimeter W + L + ˆ(w^2 + L^2). Anchor the loop at two points W units apart. Stretch the loop into a triangle so that the two sides are W and L. Trace L and W. Use the same anchor points and the same W, and make a triangle where side L is at the other end of W from the first. Trace the new L. The fourth side of the rectangle can be drawn by connecting the ends of the two L's opposite W. Extra: This method can be used to lay out other types of sport fields (soccer, basketball, football etc.) as well as a parking lot, house etc. Susan Quan Cut the string as long as the desired perimeter. Lay out the string on four sides as close to right angles as possible by sight. Use the tape measure to measure each pair of opposite sides and diagonal. Repeat until the opposite sides are equal and the diagonal is the square root of the length squared plus the width squared. Taryn H-Cabibi, Laura Parkinson You lay out one side, using the tape to get the length right, with a piece of string, extending the string past the corner of the court. You then use another piece of string as a compass to construct a perpendicular line at that corner. You measure the correct length along that side then continue the process of creating perpendicular lines until the rectangle is complete. Extra: any time you need to make a geometric figure that can be constructed. Becca Schuler The way to determine a right angle is to use the Pythagorean Theorem. It states that a2+b2=c2, or 32+42=52 (9+16=25). If one of the sides of the rectangle was 3 yards take one of the strings and make it 3 yards long and if another side was 4 yards take the string and make it 4 yards long. Measure from the endpoint of one side to the endpoint of the other side and move the strings until the measurement between the two is 5 yards. This will give you a right angle. If you do the same thing to the other side you will get a rectangle with 4 right angles. Matt Schellhaas The attachment is the sketch I created to help me solve this problem. Please relate the sketch to my following explanation. Start with a point, a, representing any corner. Make another point, b, three feet from the corner running up the length side. Measure out all points with your measuring tape. Lay out a piece of string connecting these points creating a segment ab. Next, make another point, c, four feet from the corner running along the width side. Lay a piece of string out connecting the points creating segment ac. You now a "L." Now, using the Pythagorean Theorem, fill in 3 and 4 with a and b of the theorem. Doing simple math you discover c of the theorem is 5. Lay a 5 foot piece of string out connecting points b and c. Now you have a right triangle giving you one 90 degree angle. Using 20 x 40 as example dimensions, you just form a rectangle with your string. As long as you fit your one width of 20 feet on to your four foot long side of the triangle and your 40 foot length side on to the three foot side of the triangle all the sides will fit together creating four 90 degree angles. You now have a lay out of a square tennis court with four 90 degree angles. Some other applications of this problem can be constructing houses or putting in a rectangular swimming pool. Jacob Remes, Luke Diamond, Jonas Ljung For each right angle you wish to construct: Take a piece of string that is the length of the line you wish to construct a perpendicular one to. Choose a set length (say 10 units) then construct a circle (take a piece of string and swing it around) with the center (the end of the string) 10 units ways from your vertex of your angle. Do the same ten units away in the other direction. take a fourth piece of string to and bring it from where the circles meet to where the vertex is. You now have a right angle. Dana To lay out the rectangle you can begin with a simple line. This line we will call line AB. First you measure this line with your string and tape measure. For this problem we will use the Perpendicular Bisector Theorem. This theorem states that the perpendicular bisector of a segment, in a plane, is the set of all points of the plane that are equidistant from the end points of the segment. For line AB let C be the midpoint of the line. From point C measure out a right triangle in both directions by using A^2+B^2=C^2. Once this angle is exactly 90 degrees you can measure an accurate perpendicular line from point C. This perpendicular line will be L and Point P is on Line L. From this ACP and BCP are both right angles. To complete the rectangle you will measure a parallel line to line AB and extends through point P. This line is line EF. The four corners (points) are A,B,E,F. Now connect the points A and E and points B and F. Your rectangle is now complete and all measurements that you made can be put right in to your calculations. Susan Garges You could measure two pairs of congruent lines and cut them from the string. Arrange the congruent lines parallel to one another. To position them so that the corners are right angles, you would measure a diagonal and cut it from the string. Use that diagonal of string and position the lines so that the two diagonals are equal. Once the two diagonals are congruent, you will know that the shape is a rectangle. Kim Biedermann First lay out the shape of the rectangle using the string making sure the opposite sides are congruent to each other by using the ruled tape. Then measure the two diagonals and make them congruent. Your angles are now right. If your angles were not right, the diagonal would not be equal. You could use this method to measure any convex polygon. By measuring sides to sides, angles to angles, you could be sure your opposite angles are congruent. Architects and construction workers could use this if they didn't have their materials and they needed to make sure the room or building was perfect. Jennifer Abrahamsen Since you have the dimensions, then you know the distances of the sides. Let the length = a and the width = b. If something is a rectangle, then its diagonals are equal. So first, we have to find the length of the first diagonal. We can find this using the Pythagorean Theorem, the formula for finding the hypotenuse of a right triangle. We can use this because we want to result in a right angle. So, first square a then square b and add them together. Find the square root of this sum and you will have the distance of your first diagonal. Now you have the information you need. Cut two pieces of string equal to a. Cut two pieces = b. Cut two pieces = c. Lay out a1, b1, and c1 to form a triangle. You know it is a right triangle because the only way the ends of string will meet is if they form a right triangle. Now you have laid out this figure. |\ | \ a1 | \ c1 | \ | \ ------- b1 Lay b2 and a2 to form a triangle with c1. You know it must be a right triangle because the only way the ends would meet is if they formed a right triangle. Now you have this figure. b2 -------- |\ | | \ c1 | a1 | \ | a2 | \ | | \ | -------- b1 To check that it is a rectangle all you have to do is lay c2 from the lower left corner to the upper right corner. If the string exactly fits to each point, then you have a rectangle. This is because in a rectangle the diagonals must be equal. Previous page || Next problem || Previous problem || Table of Contents || Forum Home Page