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Transcript
Math 7 – Chapter 8 Student Notes Name: _____________________________________ Teacher: ___________________________________ 1 Activating Prior Knowledge Classifying Triangles Quick Review Here are two ways to classify triangles. By side length An equilateral triangle has all sides equal. An isosceles triangle has 2 equal sides. A scalene triangle has all sides different. A right triangle has one 90° angle. An obtuse triangle has one angle greater than 90°. By angle measure An acute triangle has all angles less than 90°. Check Use square dot paper or triangular dot paper. 1. Draw an isosceles triangle. Is it acute, obtuse, or right? How do you know? 2. Draw an obtuse triangle. Is it equilateral, scalene, or isosceles? How do you know? 2 3. Can you draw an obtuse isosceles triangle? If you can, draw it. If you cannot draw the triangle, say why it cannot be drawn. 4. Can you draw a right equilateral triangle? If you can, draw it. If you cannot draw the triangle, say why it cannot be drawn. 3 Properties of a Rhombus Quick Review A rhombus is a parallelogram with all sides equal. The properties of a rhombus are: Opposite angles are equal. The diagonals intersect at right angles. Opposite sides are parallel. The diagonals bisect each other. Diagonals bisect the angles. Example Look at rhombus PQRS. a) Name all pairs of parallel sides. b) What is the measure of PCQ? How do you know? c) What do you know about the measures of QRC and SRC? d) What do you know about the lengths of SC and CQ? Solution a) Opposite sides are parallel. So, RQ is parallel to SP and SR is parallel to PQ. b) The diagonals of a rhombus intersect at right angles. So, PCQ = 90° c) The diagonals of a rhombus bisect the angles. So, QRC = SRC d) RP is the perpendicular bisector of SQ. So, SC = CQ 4 Check 5. Use a ruler and a protractor. Draw any rhombus ABCD. Draw the diagonals of the rhombus. Name: i) all equal sides ii) pairs of parallel sides iii) all equal angles iv) all right angles v) the perpendicular bisector of AC vi) the perpendicular bisector of BD 5 Parallel Lines Parallel lines are lines that never meet. They are located on the same flat 2-D surface and they are always an equal distance apart. Example: There are many ways to draw parallel lines. One method involves using a ruler and a compass. Using only a ruler and a compass, draw a set of parallel lines. STEPS 1. Draw a line segment (using a ruler). Label it AB. Mark another point, C, not on AB. 2. Mark any point D on AB. 3. Place the compass on point D. Make sure the circle drawn goes through point C. Label point E where the circle intersects AB. 4. Do not alter the compass. Place the compass on point E and draw a circle through point D. 5. Place the compass on point E and set the pencil point on C. 6. Without changing this distance, place the compass on point D and draw a circle. Where this circle intersects the other, mark point F. 7. Draw a line through C and F. CF will be parallel to AB. AB // CF 6 Example 2 Draw a line segment KL anywhere on your paper. Use a ruler to ensure that KL is 7 cm long. Construct another line segment XY so that XY // KL. You are only allowed to use a ruler and a compass. 7 Perpendicular Lines Recall that parallel lines are lines that are always an equal distance apart and they never meet or cross. Perpendicular lines are lines that meet exactly once and they intersect at right angles (they form a 90° angle). Example: As with parallel lines, there are many ways to construct perpendicular lines. You can use rulers, compasses, protractors, or even simple paper folding. We are going to use a ruler and a compass to construct two perpendicular line segments. STEPS 1. Using a ruler, draw line segment AB. Mark point C on AB. 2. Set the compass on point B and draw a circle that has a larger radius than half of the distance between B and C. (Make sure the circle intersects AB). 3. Without changing the compass’ distance, draw a second circle using point C as the center. This circle should intersect the first one in two locations. Mark these spots as point D and E. 4. Draw a line (using your ruler) through points D and E. DE is perpendicular to AB. *there are other methods of drawing perpendicular lines using a compass. 8 Example 2 Using a ruler, draw an 8 cm line segment, GH, anywhere on your paper. Create a perpendicular line segment, JK, to GH using only a ruler and a compass. 9 Constructing Perpendicular Bisectors When a line segment is divided into two equal pieces, you have bisected it. The line that divides a line segment into these equal pieces is called a bisector. If this bisector forms a right angle with the line segment, it is called a perpendicular bisector. bisector perpendicular bisector To construct a perpendicular bisector, you can use a MIRA, a ruler, or even paper folding. We are going to use a compass. STEPS: 1. Draw a line segment, AB. 2. Starting at point A, draw a circle that has a radius that is at least one-half the distance of AB. 3. Without changing the compass, draw the same circle from point B. 4. The two circles should intersect at two points. Label these points C and D. 5. Draw a line through C and D (using a ruler). CD is perpendicular to AB. Since ABCD is a rhombus (a fourequal sided 2-D shape) the diagonals (AB and CD) bisect each other so CD is the perpendicular bisector of AB. 10 Example 2 Using a ruler and a compass, draw the perpendicular bisector of a line segment, MN, that is 9cm long. Would you have to alter your process if the line segment was 1 cm long? Explain. 11 Constructing Angle Bisectors An angle is formed when two lines, line segments or rays intersect. Our goal is to learn how to divide an angle into two equal parts. The line that cuts an angle into two equal parts is called an angle bisector. STEPS: 1. Draw an angle (of any size) and label it ABC . Make sure that B is the angle’s vertex. 2. From point B draw a circle that cuts the angle in two places. Label each intersection point D and E. 3. Without changing the compass starting at point D, draw a circle (it should pass through point B) 4. Without changing the compass again, draw the same circle from point E. 5. These two circles should intersect. Label the intersection point F. 6. Connect B and F. BF is the angle bisector for angle ABC. (ABCF creates a rhombus where AC and BF are diagonals, so they bisect each other). *no matter which construction you do, you always can check your work by measuring angles with a protractor and line segments with a ruler. 12 Example 2 Using a protractor, draw an angle that measures 47° and an angle that measures 135°. Construct the angle bisector for each of the angles using only a compass and a ruler. 13 Graphing on a Coordinate Plane A coordinate plane (Cartesian Plane) is a grid system used for plotting points. You have used a portion of the plane when you have drawn bar or broken-line graphs. The coordinate plane looks like: y-axis Quadrant II Quadrant I x-axis origin Quadrant III Quadrant IV Points are always written in the form (x, y). To plot these points, you always plot the x-coordinate first. This means that you always move left or right from the origin first then up or down. Example: On the grid, plot A(3,1), B(-2,4) and C(0,-4). What is the horizontal distance between A and B? What is the vertical distance between B and C? 14 Example 2 Draw out a coordinate plane that has a maximum of +5 (for both x and y) and a minimum of –5 (for both x and y). a.) Plot the points A(4, 2), B(-3, -1), C(4, -1) and D(-3, 2). b.) Connect A to C, C to B , B to D and D to A. What shape is it? c.) Find the perimeter and area. Show your work! 15 Graphing Translations and Reflections An object that is graphed on a coordinate plane can be moved right, left, up or down (called a translation), flipped (called a reflection) or turned (called a rotation). Any movement or change to an object is generally referred to as a transformation. To complete a translation, we need to know the horizontal and vertical change. Example: On a grid, move points A(2,3), B(4,1) and (0,-1) four units left and three units down. *Notice that the original shape and its image are congruent; they are exactly the same. 16 To complete a reflection, we need to know the line in which to reflect the image. This line is called to reflection (or mirror) line. Example: Flip ABC over the x-axis if A(0,3), B(2,4) and C(1,1). Notice that the original shape and the image are congruent but have different orientations (they face different directions). 17 Example 2 Graph KLM where K(-1,-4), L(2,-1) and M(1,3). a.) Translate the triangle 2 units right and 3 units down. Label it ABC . b.) Reflect the original image over the x-axis. Label it ABC . c.) Reflect the original image over the y-axis. Label it ABC . Be sure to list the coordinates for all of the images. 18 Graphing Rotations A rotation is a turn around a point of rotation (or a turn centre). The point of rotation can be on the object or off. The rotation can be clockwise or counterclockwise. Example: Draw quadrilateral ABCD, where A(5,3), B(1, 4), C(4,1) and D(1,0). Rotate this object 90° clockwise about the point D. Notice that the original object and the image are congruent and like reflections, they have different orientations. Example: Using the shape ABCD (from above), rotate it 180° about the origin. 19 Chapter Review I should be able to: Define parallel lines, perpendicular lines, right angle, bisector, perpendicular bisector, angle, angle bisector, coordinate plane, x-axis, y-axis, origin, reflection, rotation, translation, congruent, and orientation Use a compass and a ruler to construct parallel lines, perpendicular lines, perpendicular bisectors and angle bisectors Explain in words (with or without a diagram) how to construct parallel lines, perpendicular lines, perpendicular bisectors and angle bisectors Plot points in all four quadrants of a coordinate plane Label the x-axis, y-axis, origin and quadrants on a diagram of the coordinate plane Translate, reflect, and rotate any 2-D object on a coordinate plane 20 Chapter 8: Practice Unit Exam Name: _____________ MULTIPLE CHOICE Select the best response for each of the following questions. [1 mark each] 1. The point (-2, 3) would be located in which quadrant? a. I b. II c. III d. IV 2. Which statement is FALSE? a. Parallel lines are always the same distance apart. b. A perpendicular bisector will always cut a line segment in half. c. A rotation is the only transformation that changes the orientation of a shape. d. All of these statements are true. 3. A point, P, has the coordinates (2, -8). What are its new coordinates after reflecting point P over the y-axis? a. (-2, 8) b. (-2, -8) c. (2, -8) d. (2, 8) 4. After rotating point (1,3) 90° clockwise about the origin, the new location is _________. a. (-3,1) b. (-1,3) c. (1,3) d. (3,-1) 21 5. A translation of any object changes the _____________. a. Size b. Orientation c. Location d. Congruency WRITTEN RESPONSE Read each question carefully. Show all of your work. 6. In the space provided below, draw a line segment AB that is 5 cm long. Construct another line segment, CD, so that AB // CD. Beside your illustration, explain the steps for HOW to draw the construction. [6 marks: 2 for the drawing, 4 for the explanation] 7. Add the following items to the coordinate plane below [1 mark each] a. x-axis b. y-axis c. origin d. point H(-3,1) e. point G(2, -3) 22 8. On the grid below, draw A(2, 5), B(4,0) and C(0,0). [1 mark] a.) Translate ABC 2 units left and 3 units down. Be sure to label your triangle. [3 marks] b.) Reflect ABC over the x-axis. Be sure to label your triangle. [3 marks] c.) Rotate ABC 270° clockwise about the origin. Be sure to label your triangle. [3 marks] d.) Translate ABC 1 unit right and 1 unit up and then reflect it over the yaxis. Be sure to label your triangle. [3 marks] 23 9. The picture below shows three images of an arrow. Identify each transformation. [1 mark each] Original 1. _____________ 1 2. _____________ 3 3. _____________ 2 10.Find the horizontal distance between each of the following pairs of points. [1 mark each] a. A(-5,3) and B(4, 4) b. C(3, -2) and D(0,5) 11.Find the vertical distance between each of the following pairs of points. [1 mark each] a. E(0,4) and F(6, -2) b. G(8,9) and H(-2,-5) 24 Important Terms Term Definition Illustration Triangle Equilateral triangle Isosceles triangle Scalene triangle Acute angle Right angle Obtuse angle Rhombus Congruent Parallel lines Perpendicular lines Line segment 25 Bisect Bisector Perpendicular bisector Angle bisector Coordinate grid X – axis Y – axis Origin Quadrants Compass Mira Translation Reflection 26 Rotation Prime Ordered pair 27