Measurement and Geometry (M&G3)

Transcription

1 MPM1DE Measurement and Geometry (M&G3) Please do not write in this package. Record your answers to the questions on lined paper. Make notes on new definitions such as midpoint, median, midsegment and any definitions that you need to review such as interior angle, exterior angle and reflex angle. These labs have been created to cover the following Overall Expectation from Measurement and Geometry: - Verify, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems. Describe the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons, and apply the results to problems involving the angles of polygons. Determine the relationship between the sum of the interior angles of a polygon and the number of sides. Describe some properties of polygons (related to): Midpoints of the sides of a quadrilateral, the diagonals of rectangles, the line segment joining the midpoints of two sides of a triangle Pose questions about geometric relationships, investigate them, and present their findings, using a variety of mathematical forms. Illustrate a statement about a geometric property by demonstrating the statement with multiple examples, or deny the statement on the basis of a counter-example. Lab 1: Introduction to GeoGebra & Investigation of Polygons and Angles Open GeoGebra Under Perspectives, choose Geometry Your screen should look like a gridded paper.

2 Choose the tool Polygon from the toolbar. Then click on the graphics view the gridded paper in 3 spots to create the vertices A, B, C and then click on point A again to close the triangle. You should have created a triangle now! Also it is handy to note that whenever you drag the vertices into any intersections in the gridded line, it will snap in place. This will be handy if you ever need to use this to make slopes, right angle triangles, segments with a set distance...etc. You can drag around A, B, or C to resize the triangle. Let s set up some measurements so we can see the changes as we apply them. First, let s add some angles. Click on angle. You can either select two adjacent sides, or three vertices, in order to specify an angle. You can also add distances and lengths to your triangle.

3 An example would look like this. Play around with this. See what happens when you drag around the vertices. Notice the measurements change as the size of the triangle changes. Note: If you click on the line segments (or vertices) clockwise, you will get the interior angle. In the above example, if you click on line, and then line, it will calculate the interior angle. If you click on the line segments (or vertices) counter clockwise ( then ), then it will calculate the reflex angle (angle greater than 180) otherwise known as the angle explementary to the interior angle. We are interested in finding out the sum of the angles of a triangle using this program. Before we can actually do this, we need other functions to be visible from GeoGebra! Select the Algebra option under View. Also select the Input Bar and select Show. You should now see this on your screen, with everything displayed on your left.

4 On the input bar below, let s establish our SUM calculations. Take a look at what your angles are called on the left hand column. If you followed this guide so far without making additional random points, they should be. (Since they go in order in the Greek alphabet) You can select the Greek letters by clicking on the button on the right side of the input bar. Input the formula in the picture. The SUM= in the beginning sets the name for the output. The formula α + β + γ is calculating the sum of the interior angles. Now that we have this setup, what can we investigate and explore? You now have the ability to label distances, angles, and sums of interior angles so you should be able to more easily investigate the angle relationships of polygons Q1 What is the sum of interior angles for other polygons? 4-sided? 5-sided? 6-sided? Draw the shapes and create a formula to have GeoGebra sum up the angles of each polygon. *note* make a new worksheet if you are making a new shape. It will reset the Greek letters to begin with. From your results, can you predict what the sum of the interior angles of a 10 sided polygon would be? A 100-sided polygon? Q2. A diagonal of a polygon is a segment joining 2 non-consecutive vertices of a polygon. How many diagonals can be drawn from one vertex of a 4-sided polygon? What shapes are formed? How many diagonals can be drawn from one vertex of a 5-sided polygon? 6-sided? Use GeoGebra to investigate how we find the number of diagonals that can be drawn from one vertex of an n-sided polygon without having to count all the possible diagonals.

5 Q3. Consider your answers to questions 1 and 2. Try to find a relationship between the number of sides of a polygon and the sum of the interior angles. Explain this relationship. It might be useful to organize your data in a table to make connections. Number of Sides Number of Diagonals Number of Triangles Created Sum of the Interior Angles Q4. An exterior angle of a polygon is created by extending a side of the polygon. NB: An exterior angle and a reflex angle are NOT the same thing. Exterior angles are never greater than 180 degrees. The exterior angle will be adjacent and supplementary to one of the interior angles of your polygon. Create a triangle. Create and label the measures of the 3 exterior angles of your triangle. Using the skills you used in question 1, find the sum of the exterior angles of your triangle. Will this sum be the same for all triangles? Q5. Investigate the sum of the exterior angles of polygons other than triangles. What can you conclude?

6 Lab 2: Midpoints, Medians, and Midsegments Let s get you familiar with a few additional skills before investigating other properties of triangles. Area of a polygon Create a polygon according to what you learned last time. Make sure it s set up so you have the Algebra screen on the left and the Input bar on the bottom. Type in formulas for the area in the input bar. Remember to name it something you will be able to identify. Here we will name it Area. Now to put in the formula itself. Midpoint of a line segment Make sure you have your shape Select the Midpoint or Center option

7 Click on the segment that you want a midpoint, and voila, You have a midpoint! Creating a line segment Create 2 new points. Choose the Segment between Two Points, and click on the two points that you ve created. You should now have a line segment! Q1. A median of a triangle is a segment joining the vertex of a triangle to the midpoint of the opposite side of a triangle. Create a triangle and use the skills that you have learned to create medians. How many medians does a triangle have? What do you notice about the medians? The intersection of the medians is called a centroid. Determine whether the centroid is always inside the triangle using GeoGebra.

8 Q2. A midsegment of a triangle is a segment joining the midpoints of 2 sides of a triangle. Create a triangle and use the skills that you have learned to create a midsegment. What do you think is true about a midsegment of a triangle? Construct all of the midsegments of a triangle. Write down any observations that you make. Q3. A midsegment of a quadrilateral is a segment joining the midpoints of consecutive sides of the quadrilateral. Create a quadrilateral and use the skills that you have learned to create the four midsegments of your quadrilateral. Find the angle measures and side lengths of the new shape created. What shape is created? Will it always create this shape? Verify by dragging the vertices of the original quadrilateral. Lab 3: Illustrate a statement about a geometric property by demonstrating the statement with multiple examples, or deny the statement on the basis of a counter-example. Use GeoGebra to investigate the validity of the following statements: Q1. The diagonals of a rectangle are congruent (the same length). Q2. The diagonals of a rectangle are perpendicular. Q3. The diagonals of a square are perpendicular. Q4. The diagonals of a rhombus bisect the angles to which they are drawn. Q5. The median of a triangle cuts the triangle into 2 triangles of equal area. Q6. The diagonals of a rectangle intersect to create 4 congruent triangles. Q7. The opposite angles of a parallelogram are supplementary (add to 180). Q8. The midsegments of a square create a square with area half of the original square. Q9. The diagonals of a parallelogram bisect each other. Q10. No angles of an isosceles triangle are congruent. Q11. Consecutive angles of a parallelogram are supplementary. Q12. The sum of 2 sides of a triangle is never equal to the third side.