Geometry And You (1948)

[Music] this is the new house to be built for the Wilkins family Henry and Walter have been excited about the model ever since the architect delivered it the other night Henry the artist in the family likes the beauty in proportion Walter who wants to be an engineer admires the balance and the massive appearance as a matter of fact it is the geometry of the design that pleases both of them the problem right now is a Terrace they want to add a porch over the East end of it by enclosing this part of the Terrace the family will have a cozy outdoor spot in which to relax how will the porch change the Terrace and the appearance of the house though and what about those big windows on the south mother wants to know will the porch cut off the winter sun that will help warm the living room we wouldn't want to do that and there's one way to be sure let's build the porch here change the model to match the blueprint and try it out the geometry that helped build the model will help us add to it for our geometric construction we'll need some construction tools a set of compasses and a straight edge we'll also use measuring tools a protractor and a scale now let's see how large is the model 38 of an inch on the model represents 1T on the actual house and the blueprints 1/4 in represents 1T so 38 compared to 1/4 is how much it looks as though the dimensions of the model are just one and 1/2 times the corresponding dimensions of the blueprints to build the porch we'll have to construct similar rectangles walls the same shape as those on the plan but with larger Dimensions one and 1/ half times as large so before long the porch begins to take shape now we're concerned with only two dimensions for each side length and width we don't bother with the thickness of the cardboard and we speak of the walls as plain f figures considering them to be perfectly flat since these plain figures are drawn to scale and have the same shape as the plain figures on the blueprint we shall call them similar figures for the triangle of the Gable in this print we'll have to make another similar figure a triangle one and 1/2 times as long and as high as this one first a straight line the base of the triangle only one and 1/ half times as long as the base on the blueprint now to measure an angle line up the protractor with a center point at the vertex of the angle line up the zero with the initial side of the angle all right there's a 20° Angle now 20° but what is Henry doing 30° it looks as though he's making the angle one and 1 half times as big as the one on the blueprint what kind of a triangle do you think he's going to get out of that doesn't match the blueprint does it Henry no angles are not drawn to scale to make similar triangles that is triangles of the same shape their corresponding angles must be equal the size of the angles always determines the shape of the triangle so Walter marks off 20° angles on the cardboard and completes the triangle by extending the sides of these angles until they meet remember the geometric principle of similar triangles it has been the same for for 23 centuries since uid a Greek mathematician determined some of the principles of the mathematical science of geometry if two angles of one triangle are respectively equal to two angles of another triangle the triangles are similar the base of one is 1 and 1/2 times as long as the base of the other so the other sides are also in the ratio of 1 and 1/2 to 1 these are similar figures now to finish we add drawings on paper of the porch screens and that was easy to do just match each wall with a cover the same size and shape but this East wall with a triangular Gable gives trouble we need a triangle that just matches it and it's hard to measure length up under the Gable well there's another geometric principle that will help us here how does the theorem go if two angles and the included side of one triangle are respectively equal to two angles and the included side of another the triangles are congruent the figures have not only the same shape but the same size and all their angles and sides are respectively equal so we make a line equal to the base line of the Gable and with the protractor we lay out 20° angles equal to the angles of the Gable this will make a triangle congruent to the triangle of the Gable all sides and angles respectively equal this is only one of the ways we can make congruent triangles when two figures are congruent it means that one can be placed over the other so that all the respective parts will coincide well geometric construction has enabled us to build the porch to the same scale as the rest of the house and the porch does look attractive doesn't it it fits in with the lines of the house parallel lines rectangles geometric shapes proportions yes it takes those things to make a beautiful house but wasn't there something about symmetry that's a geometric idea too isn't it the main features on one side of this line are matched on the other side symmetry is an exact balance of shapes and lines on each side of a central axis and each window is symmetrical CLE 2 balanced about a Center Line called the axis of symmetry if you look around you you'll recognize geometric forms in many buildings in a large office building we see rectangles parallel lines and perpendiculars they are there because an architect used geometry in his plans just as Engineers who lay out great railroads and highways use geometry in their work so too do the Builders of locomotives Automobiles and Machines of all kinds what modern machine could do its work without the circle look at the geometric Precision of the printing press whose wheels and cylinders bring us News of the World about us a world of geometric shapes in steel concrete and in nature for we even use geometry to determine the pattern of these fields through a geometric Arc the Sun appears to move over the Earth over our homes and what was it Henry and Walter wanted to know will the new PCH reduce the amount of warming winter sunlight how does the sun move in this pamphlet Walter has the geometric answer where they live the Winter Sun reaches a peak of at noon of about 30° above the Horizon now let's arrange a model Sun for our model house this flood lamp will do we'll move it through an arc similar to The Arc of the sun it should rise in the east like this to a noon position in line with the 30° triangle now let's see how much Light reaches the living room windows we can see that the full afternoon sun will warm the living room through the windows in the winter yes geometry helped build the porch and proved that it was practical no wonder Henry and Walter are eager to learn more about a mathematical science that is so important to all of us the science of geometry