Transcript Math SCO E8
E8 Students are expected to make generalizations about the diagonal properties of squares and rectangles and apply these properties What do we mean by the ‘diagonal’ of a two-dimensional figure? • Use a Mathematics Dictionary to find the definition for the term ‘diagonal’. Record this definition. Investigate the diagonals of squares by completing the activities below for each of the squares given to you. •How do the lengths of the two diagonals of your squares compare? •Check by measuring them. Share your findings. •How do the diagonals of the squares appear to intersect? •Check your prediction by measuring. Share your findings. •What do the diagonals appear to do to each vertex angle? •What do you think the measure of each of the angles at a vertex is? •Check by measuring. Share your findings. •Next, cut out the four triangles made by the two diagonals. Describe and compare these triangles. What did you learn about the diagonals of a square during this investigation? The Diagonals of a Square: •are equal in length •bisect each other •intersect to form four right angles and combined with the previous properties this means they are perpendicular-bisectors of each other •are bisectors of the vertex angles of the square, thus forming 45 degree angles •form four congruent isosceles right triangles Investigate the diagonals of squares by completing the activities below for each of the squares given to you. •How do the lengths of the two diagonals of the rectangle compare? •Check by measuring them. Share your findings. •How do the diagonals of the rectangle appear to intersect? •Check your prediction by measuring. Share your findings. •What do the diagonals appear to do to each vertex angle? •What do you think the measure of each of the angles at a vertex is? •Check by measuring. Share your findings. •Next, cut out the four triangles made by the two diagonals. Describe and compare these triangles. What did you learn about the diagonals of a rectangle during this investigation? The Diagonals of a Rectangle: •are equal in length •bisect each other •form two pairs of equal opposite angles at the point of intersection •form two angles at each vertex of the rectangle that sum to 90 degrees and have the same measures as the two angles at the other vertices •form two pairs of congruent isosceles triangles • E8.1 Draw squares that have diagonals of length 8cm. What properties of a square did you use to do this? Did everyone draw the same square? • E8.2 Cut a square along both diagonals. Investigate the different shapes you can make (i) using two of the triangles formed if equal sides must abut, (ii) using three of the triangles, and (iii) using all four triangles. • E8.3 Draw rectangles that have diagonals that intersect to form 60 degree angles. Did everyone in the class draw the same one? How do all the rectangles compare? • E8.4 When the diagonals are drawn in a rectangle, how do you know that each triangle formed is ¼ of the rectangle? • E8.5 Draw an isosceles right triangle. Use a mira to draw the square for which the triangle is one-quarter. • E8.6 Draw a segment 12cm long. Use only a mira to construct the square that has this as a diagonal. • E8.7 All triangles are rigid while rectangles are not. One or both diagonals are often used in the real world to make a rectangular shape rigid. Explain what this means and give a real world example. • E8.8 Draw isosceles triangles. Explain how you could use these triangles to construct rectangles that would have the isosceles triangles represent one-quarter of their areas. E8.9 Imagine a family of rectangles has a perimeter of 38cm and all of their sides are whole numbers. Draw this family of rectangles on graph paper. Which family member has the greatest area? The longest diagonal? • E8.10 Draw rectangles and show the two diagonals. Measure one angle at a vertex and one angle at the centre. Then find the measures of all the other angles in the figure using only these two angle measurements and your knowledge of properties.