Why brace the gate? - January 2-5, 1996
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I have two gates in my yard. One looks like the picture below. What purpose does the diagonal brace serve? Why can't the gate just be a rectangle? Convince me that your answer is right - my neighbor has been asking a lot of questions lately and I want to have a good explanation for him.
I am in the middle of building another gate, and there is a cable running on the diagonal of the gate. I want to know if it's square so that I can attach the cable, but I've managed to lose all my tape measures, and all I've got is a roll of string. I know that the opposite sides are equal because I measured them carefully last week, before I lost my tape measures. How can I make sure that all the angles are 90 degrees?
Annie says: Solutions
There were lots of wrong answers this week, and the reason was that I was really tough in 'grading' the first part of the question about the purpose of the diagonal brace. A lot of folks wrote that it was for 'support' without explaining why a diagonal brace, as opposed to vertical or horizontal, is actually useful.
I finally decided to accept answers that mentioned support in conjunction with triangles, or answers that mentioned rectangles and parallelograms or at least rectangles sagging when weight is applied opposite the hinges. It was a tough line to draw between right and wrong in some instances! It is important, however, to really explain your reasoning and to be rigorous not just in 'what' but in 'why'. Saying that the diagonal keeps the rectangle from breaking or falling apart did not cut it.
A lot of people did some extra work on the second part, which is partly my fault. I asked how I could tell if the gate was 'square', by which I meant that all angles are 90 degrees. Many of you thought I meant 'a square', as in all sides equal, and showed ways to show that all the sides were the same length. While I gave these folks credit, you need to read the problem carefully, especially if you have any questions at all. The problem states that the opposite sides of my gate are equal, which is enough to continue. Read carefully and save yourself some work.
Following are highlights, and the names of people who submitted correct solutions and most of the solutions are also available. I think that a great many of the 'wrong' answers were really correct, in the sense that the writer understood what the diagonal was doing there, but just didn't say so. Read these solutions and see what you think of them.
Annie McIntyre and Lauren Wall Grade 9 Mount St. Joseph's Academy, Flourtown, PA The diagonal brace on your gate forms two triangles. By the side-side-side postulate, a triangle cannot change its angles without changing the lengths of its sides. On the other hand, a quadrilateral can change its angles without changing its sides. Therefore, the triangles formed by the brace are more stable than the quadrilateral gate, because the sides of the triangles cannot change and neither can the angles. To see if your other gate is a square, take your piece of string and make it a diagonal of your gate and then use it to make a second diagonal. If it takes the same amount of string to make both diagonals, then the diagonals bisect each other. Your gate then must be a rectangle or a square with 90 degree angles. To see if it's a square, measure the sides with your piece of string, and if all sides use the same amount of string, it is a square.
Shem Thompson Grade: 9 School: Newport High School Teacher: Mr. Mabbott The reason the diagonal is there is for engineering reasons - triangles are stronger than quadrilaterals and with the diagonal the quadrilateral becomes two triangles. For geometric reasons it's to keep angles square by keeping the opposite angles from moving away from or closer to each other, squashing the figure, and making the angle obtuse or acute. To answer your second question, take the string and lay it across the diagonal opposite the diagonal with the cable across it. Move the corners of the quadrilateral until the length of the string is the same length as the length of the cable. To make sure the length of the string and cable are the same just pick up the string and put it next to the cable to compare; then put the string back. The reason I am doing this is because the two quadrilaterals with square angles (square and rectangle) are the only two quadrilaterals with diagonals equal in length.
Kristy Giballa Grade 10 Mount Saint Joseph Academy, Flourtown, PA To answer your first question, you can tell your neighbor that the the diagonal brace divides the gate into two triangles for support. A triangle is more supportive because the lengths of the sides have to change before the angles can. In a quadrilateral if you move the sides the angles change (if someone sat on the gate it would slant). Since there is not much chance of the sides of your gate changing length the angles won't move it if they are parts of triangles. Also, by the Side Side Side postulate (if three sides of one triangle are congruent to three sides of another triangle then the triangles are congruent) the two triangles are equal because of the brace and hold up perfectly. To answer your second question about the other gate, I'll tell you what you can do until you find a tape measure. You already know that the gate is a parallelogram because both pairs of opposite sides are congruent. Use the string to measure the lengths of the adjacent sides. If they are the same it is a rhombus. Then if the diagonals are congruent it is a rectangle. If you combine what you found out and conclude that it is a square with all 90 degree angles (four equal sides and four 90 degree angles).
∠Colleen Cusick Grade 10 Mt. St. Joseph Academy, Flourtown, PA This week's problem of the week was to explain why a gate has a diagonal brace on it and to figure out how to prove that a gate is a square using only a roll of string. The first part was easier. The gate was cut by its diagonal so that it could be divided into triangles. A triangle is very sturdy and very desirable in architecture because of the SSS postulate. Given the lengths of each side of the triangle there is only one way in which they could meet. There is only one value that each angle could be. The triangle could not bend or flatten, as a quadrilateral can, without breaking or changing the measure of at least one side, so the gate will be stronger and sturdier. You can measure the angles of the gate and prove that it is a square using only the roll of string very easily. Because the gate was measured earlier you have for your given information that it is a parallelogram. You can use the string to prove that the parallelogram is a rhombus by putting the string against one side of the gate, marking off where the gate ends, and then putting it against a consecutive side. If the sides match both ends at the same mark on the string, then the figure is a rhombus (if consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus). To prove that the angles are all 90 degrees and that the figure is a rectangle, you place the string on one of the diagonals and mark off how long it is. Then you place it on the other diagonal. If the diagonals are the same length, then the parallelogram is a rectangle (the diagonals of a rectangle are congruent). If the gate is both a rectangle and a rhombus, then it is a square.
Carly Tubbs Grade 9 Fairfield High School, Fairfield, Connecticut The diagonal brace runs from one corner to the opposite corner, dividing the rectangle into two congruent triangles, congruent by SSS. The triangles are polygons of closed figures in a plane that is made up of sides that intersect only at vertices. Triangles are the most rigid polygons because they have only 3 vertices, the least number possible. This allows each side of the triangle to push against each other, one on each side, and support itself. It is also the strongest shape because there are only 3 vertices, which is the weakest part of a polygon. By considering the 2 triangles as separate, strong polygons, then they come together to form a rectangle that is stronger as a result of the support given by the brace. It can't just be a rectangle because each side does not push against all other sides, and has 4 weak vertices, giving it leeway to wobble. You can make sure it is square by using the string to measure from one corner to the opposite corner. Then take another piece of string and measure from the 3rd corner to the opposite one. If the 2 pieces are equal, then the angles are 90 degrees. This is because of the theorem where if the diagonals of a parallelogram are congruent, then it is a rectangle. As you already know that it is a parallelogram because you measured the wood, then only the diagonals have to be congruent to prove the rest of the definition of a rectangle - that all angles are 90 degrees.
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