SS - Circle through 3 points Drawing a circle through three given points with compass and straightedge 1. Draw the only circle that passes through the three points below. 2. Draw the only circle that passes through the three points below. (C) Copyright John Page 2007 SS - Constructions Introduction to constructions Constructions: The drawing of various shapes using only a pair of compasses and straightedge or ruler. No measurement of lengths or angles is allowed. The word construction in geometry has a very specific meaning: the drawing of geometric items such as lines and circles using only compasses and straightedge or ruler. Compasses Compasses are a drawing instrument used for drawing circles and arcs. This kind of compass has nothing to do with the kind used find the north direction when you are lost. Straightedge A straightedge is simply a guide for the pencil when drawing straight lines. Why we learn about constructions The Greeks formulated much of what we think of as geometry over 2000 years ago. Why did Euclid do it this way? Why didn't Euclid just measure things with a ruler and calculate lengths? One theory is the the Greeks could not easily do arithmetic. To find out more Constructions pages on this site Lines Angles Triangles Right triangles Triangle Centers
SS - Rotational Symmetry Maths Rotational Symmetry Symmetry means balance or form. In maths we often talk about shapes and things being symmetrical. There are two types of symmetry, line symmetry and rotational symmetry. Rotational symmetry If we turn an object round will it look the same? Here is an example We have put a blob in one corner to show it turning round. You see that apart from the blob the shape looks exactly the same in 1 and 3. Here is a letter with rotational order of two. You could turn (rotate) the letter s around to its new position and you would not know it had changed (we have put the blob on to show you). What do you think the rotational symmetry order of A is? Answer Answer A has got rotational symmetry of order 1 This is just a complicated why of saying that you can not turn A around to any other position so it looks the same. So rotational symmetry order of 1 means NO rotational symmetry (we can't rotate it). Try and find out the rotational order of symmetry of the following shapes and letters.
SS - Congruent Triangles Congruent triangles are a special type of similar triangles. Congruent triangles have the same shape (similar triangles) and size. Increase/decrease <A, <B and <C by clicking and dragging the vertices of the left triangle below. Notice the corresponding angles (<D <E and <F) remain congruent. Since the sum of three angles must be 180o, the third pair of corresponding angles must also be congruent when the first two pairs of corresponding angles are congruent. In the applet above: Notice the size and shape of the new pairs of triangles remains the same. Two triangles are congruent if: all 6 pairs of corresponding angles and sides are congruent. The factor for congruent triangles is 1. If all 3 pairs of corresponding sides in two triangles are the same, then the triangles are congruent. Move the corners A, B or C of the triangle above.
SS - Platonic Realms Home Page Sally Simplewit's mother has been rearranging the dining room again, but she is frustrated; the place she wants to put the dining room table isn't working out, because the floor there is smooth but uneven, and she can't seem to get the table into a position where it doesn't rock. It is a round table, with four legs of equal length. Frustrated, she announces that she is going to get a saw and shorten one of the table's legs. "For," says Sally, "I guarantee I can get all four legs firmly on the floor without changing either the floor or the lengths of the legs!" Today's challenge: How can Sally make such a guarantee? Solution to yesterday's challenge THE MATH STORE LIVES! NEW FOR 2012: Platonic Realms is at last being rewritten and moved to a new server. This site is registered with the Internet Content Rating Association (ICRA), and is suitable for all ages.
SS - Similar Triangles Definition: Triangles are similar if they have the same shape, but can be different sizes. (They are still similar even if one is rotated, or one is a mirror image of the other). Try this Drag any orange dot at either triangle's vertex. Both triangles will change shape and remain similar to each other. Triangles are similar if they have the same shape, but not necessarily the same size. which is read as "Triangle PQR is similar to triangle P'Q'R' ". Properties of Similar Triangles Corresponding angles are congruent (same measure)So in the figure above, the angle P=P', Q=Q', and R=R'. Rotation One triangle can be rotated, but as long as they are the same shape, the triangles are still similar. In this particular example, the triangles are the same size, so they are also congruent. Reflection One triangle can be a mirror image of the other, but as long as they are the same shape, the triangles are still similar. How to tell if triangles are similar Similar Triangles can have shared parts
SS - Congruent Triangles Definition: Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other. In the simple case below, the two triangles PQR and LMN are congruent because every corresponding side has the same length, and every corresponding angle has the same measure. The angle at P has the same measure (in degrees) as the angle at L, the side PQ is the same length as the side LM etc. Try this Drag any orange dot at P,Q,R. In the diagram above, the triangles are drawn next to each other and it is obvious they are identical. Imagine the triangles are cardboard One way to think about triangle congruence is to imagine they are made of cardboard. How to tell if triangles are congruent Any triangle is defined by six measures (three sides, three angles). AAA does not work. They are called similar triangles (See Similar Triangles). SSA does not work. Constructions Properties of Congruent Triangles
SS - Similar Triangles Applications Image Source: A powerful Zoom lens for a 35mm camera can be very expensive, because it actually contains a number of highly precise glass lenses, which need to be moved by a tiny motor into very exact positions as the camera auto focuses. The Geometry and Mathematics of these lenses is very involved, and they cannot be simply mass produced and tested by computer robots. Lots of effort required to manufacture these lenses results in their very high price tags. Here is a diagram showing how the zoom lens internal arrangement changes as we zoom from 18mmm wide angle to 200mm fully zoomed in: Image Source: Image Copyright 2013 by Passy’s World of Mathematics Shown above are some band photographs taken by Passy with a special low light camera. Unfortunately this camera does not have a zoom lens, and so you need to be right up close to the stage to take good pictures. Measuring heights of tall objects is also covered in this lesson. Bow Tie Triangles
SS - Similar Triangles Definitions and Problems Definition Generally, two triangles are said to be similar if they have the same shape, even if they are scaled, rotated or even flipped over. The mathematical presentation of two similar triangles A1B1C1 and A2B2C2 as shown by the figure beside is: Two triangles are similar if: 1. Each angle in one triangle is congruent with (equal to) its corresponding angle in the other triangle i.e.: ∠A1 = ∠A2, ∠B1 = ∠B2 and ∠C1 = ∠C2 2. 3. Be careful not to mix similar triangles with identical triangle. Therefore, all identical triangles are similar. Although the above shows that we need to know the measures of the three angles or the lengths of the three sides of each triangle in order to decide whether the two triangles are similar or not, it would be sufficient, for solving problems involving similar triangles, to know only three of the above measures for each triangle. 1) the three angles of each triangle (without the need to know the lengths of their sides). Solution: Practical Examples