Slide 2 When we look carefully at bridges, we can see how structural engineers use different shapes to make the overall design. We can see triangles and squares. We can even see parabolas. Slide 3 Structural engineers use the same types of shapes in buildings. Many building frames are simply repeating squares, as shown in the top left. The bottom left image shows how a square is reinforced by adding a diagonal cross brace in this scaffolding, which breaks the square into two triangles.
The image on the right shows an Antarctic geodesic under construction. The structure of geodesic domes is similar to the structure of soccer balls and can be viewed as a group of pentagons and hexagons. But, if we break each of those shapes down, we can see that they are fundamentally composed of triangles.
Slide 4 Even when we get outside the realm of civil or architectural engineering, we can see how engineers rely on the known strength of shapes. A motorcycle frame uses many triangles to support the wheels and seats.
Mechanical engineers design cranes, which use triangles and squares in their frames. Even satellites use these familiar and basic regular geometries. Slide 5 On your paper, sketch each of these regular polygons: square, diamond and triangle. If we push straight down on a shape, putting the whole shape into compression, what happens to the shape?
Draw, using a different pen or pencil or dashed line, how the shape would look if you pushed on it. Assume that the sides of the shape are rigid and won't change length or bend. Slide 6 Take a look at this! If you push down on top of the square, it will no longer be a square, but instead takes the shape of a rhombus, which is a type of parallelogram.
This is called "racking. But what about the triangle? The triangle maintains its shape! Slide 7 The reason that the square and diamond collapse is because the angle between the structural members can change without having the length of the members change or bend. Remember back to geometry when we talked about how polygons are defined? In this case, both quadrilaterals simply require the sum of the interior angles to equal degrees, but each angle can change.
Slide 8 Triangles are unique in that sense. The angle between two sides of the triangle is based on the length of the opposite side of the triangle. Do you remember this from geometry? The angle "a" is fixed, based on the relative length of side "A. Slide 9 As we showed, other regular polygons can be deformed without changing the length of the sides.
A square loses its shape as its right angles collapse, and a pentagon and hexagon can be deformed. But the shapes stay "closed" because the sum of the interior angles is kept constant. So what can we do to the other shapes, the squares, pentagons and hexagons, to keep them from collapsing?
Draw these shapes on your paper and add what would be necessary. Slide 10 Did you break the shapes into triangles? Since we know a triangle cannot collapse, and we know that these regular polygons can always be reduced to triangles that's how we figure out the sum of the interior angles, remember? Slide 11 The same concept applies in three dimensions. As shown, a cube can collapse by "racking," just like the square we saw collapse in two dimensions.
So what would we do to make a strong 3D structure? Slide 12 We make 3D triangles! Specifically, we can make rectangular or triangular pyramids!
This is why structural engineers rely on triangles, both in 2D and in 3D, to make strong structures! A 3D structure made of individual structural triangles like this is called a "truss," and is used throughout engineering for a strong light-weight structure! Now that we have reviewed the basics of how structural engineers rely on structural shapes, I expect that you will start to notice in your day-to-day life the way in which things are built.
Look around you, at the buildings, and cranes, and bridges, and houses, and cars, and furniture, and you will see that so much of structural engineering is based upon these fundamental and simple shapes. Next, in the activity, you will experience designing, building and testing structural trusses. So keep in mind the discussions we've had about the different shapes and how they can be used to make strong structures.
For example to enable workers to lay brick, install trim or paint. Opening Question: Ask students what regular geometry triangle, square, circle, pentagon, hexagon, etc. Answer: A triangle is the strongest shape, and in this lesson, we will find out why! Triangle shapes are commonly used for strength and structural support. Carlson, University of Colorado Boulder. Embedded Geometry Practice: Have students participate by completing the two drawing tasks outlined in the PowerPoint presentation.
Personal Relevance: After presenting the PowerPoint slides or as a homework assignment , have students individually list on their papers the places, objects, structures and products where they have seen triangles functioning as structural shapes. After five minutes, have each student read their list to the class while you compile a master list of their responses on the board. Possible answers: Bridges, transmission towers, cranes, peaked roofs, tables, chairs, bicycles, bike racks, railings, fences, gates, shelf supports, brackets, billboard sign supports, etc.
If the measures of corresponding sides are known, then their proportionality can be calculated. If all three pairs are in proportion, then the triangles are similar. Similar triangles are easy to identify because you can apply three theorems specific to triangles. Explanation: Right triangles are congruent if both the hypotenuse and one leg are the same length.
These triangles are congruent by HL, or hypotenuse-leg. AA — where two of the angles are same. As the two sides of a triangle comparing to the corresponding sides in the other are in same proportion, and the angle in the middle are equal, the above triangles are similar, with the prove of SAS. Therefore, the answer is C. For two rectangles to be similar, their sides have to be proportional form equal ratios. The ratio of the two longer sides should equal the ratio of the two shorter sides.
A parallelogram has adjacent sides with the lengths of and. Find a pair of possible adjacent side lengths for a similar parallelogram. Explanation: Since the two parallelogram are similar, each of the corresponding sides must have the same ratio. Two quadrilaterals are similar quadrilaterals when the three corresponding angles are the same the fourth angles automatically become the same as the interior angle sum is degrees , and two adjacent sides have equal ratios.
Now, all squares are always similar. Their size may not be equal but their ratios of corresponding parts will always be equal. As, the ratio of their corresponding sides is equal hence, the two squares are similar. Begin typing your search term above and press enter to search. Press ESC to cancel. Skip to content Home Articles Why are triangles the best shape for bridges? Ben Davis April 22, Why are triangles the best shape for bridges?
Why are congruent triangles used in construction of bridges and buildings? Why are triangles chosen in the design of trusses instead of squares? What are the advantages of using similar triangles in the construction? What is the importance of similar triangles? What are the rules for similar triangles? How can we apply our knowledge in congruent triangles in our daily lives? How do you prove similar triangles? What are the 3 ways to prove triangles are similar?
0コメント