Carefully construct the three medians of the second triangle.Carefully construct the three altitudes of the first triangle.Use one triangle for each construction below. And remember that the altitude may fall outside the triangle, so you might want to draw or fold an extension of the sides of the triangle to help you.ĭraw five triangles, each on its own piece of patty paper. When you construct altitudes, you need to construct a perpendicular to a segment, but not necessarily at the midpoint of that segment. Except in the case of special triangles (such as an equilateral triangle, and one median in an isosceles triangle), you can’t construct a median with just one fold. When you construct medians, you need to do two things: First find the midpoint then fold or draw a segment connecting that point to the opposite vertex. The intersection of the crease and the original line segment is the midpoint of the line segment. Next, fold the paper so that the endpoints of the line segment overlap. To construct the midpoint of a line segment, start by drawing a line segment on the patty paper. Here is a sample construction with patty paper to get you started: Throughout this part of the session, use just a pen or pencil, your straightedge, and patty paper to complete the constructions described in the problems. Though your “straightedge” might actually be a ruler, don’t measure! Use it only to draw straight segments. Since you can see through the paper, you can use the folds to create geometric objects. You can fold the patty paper to create creases. In the problems below, your tools will be a straightedge and patty paper. The most common tools for constructions in geometry are a straightedge (a ruler without any markings on it) and a compass (used for drawing circles). A construction is a method, while a picture merely illustrates the method. It shows how a figure can be accurately drawn with a specified set of tools. The essential element of a construction is that it is a kind of guaranteed recipe. Drawings are intended to aid memory, thinking, or communication, and they needn’t be much more than rough sketches to serve this purpose quite well. Go to and search about compass and straightedge construction.Geometers distinguish between a drawing and a construction. Using compass and straightedge, what kind of line segments can we construct?Ħ.
Use compass and straightedge to constructĥ. Given three line segments of length $r$, $r a$ and $r b$. Use trigonometry to prove the Right Triangle Altitude Theorem.Ĥ. Write about your favourite example of similar triangles.ģ. Prove that if two triangles have two pairs of equal angles then all their three pairs of angles are equal.Ģ. We will see the reason why the Right Triangle Altitude Theorem makes it possible for this construction. Hope to see you again then.ġ. In the next post, we will explore Gauss' construction of a regular 17-polygon. So indeed, the big square $CAIJ$ has area equal to the sum of two smaller squares $ABXY$ and $BCPQ$, and we have obtained the Pythagorean Theorem. The right identity $CB^2 = CH \times CA = CH \times CJ$ shows that the square $BCPQ$ has the same area as the rectangle $CHMJ$.Using the left identity $AB^2 = AH \times AC = AH \times AI$, we can see that the square $ABXY$ has the same area as the rectangle $AHMI$.Pythagorean Theorem says that the two squares $ABXY$ and $BCPQ$ have a total area equal to the big square $CAIJ$.