2. Use only a compass and a straightedge

2. Use only a compass and a straightedge

2. Use only a compass and a straightedge to construct each of the following, if possible: A segment congruent to and an angle congruent CB to ∠ACB C A B b. A triangle with sides of lengths 2, 3, and 4 units c. An isosceles right triangle d. An equilateral triangle with any side length 4. A rural homeowner had his television antenna held in place by three guy wires, as shown in the following figure. a. If the stakes are on level ground and the distances from the stakes to the base of the antenna are the same, what is true about the lengths of the wires? Why? b. If the stakes are not on level ground yet are the same distance from the base of the antenna, explain whether you can make the same conclusion regarding the lengths of the wires. a. if the ground is level, and the height of the mast is a common side to all three triangles you’ve got side/angle/side congruent triangles so the wire lengths are the same. b. if the ground is not level then the lengths might differ because that angle differs. Assessment 12-2B: Exercise 1 1. Use any tools to construct each of the following, if possible: a. A right triangle with one acute angle measuring 75° and a leg of 5 cm on a side of the 75° angle b. A triangle with angles measuring 30°, 60°, and 90° Assessment 12-3B Exercises 11 & 14 11. In the parallelogram shown, find x in terms of a and b. 14. Using any tools, construct each of the following, if possible. If the construction is not possible, explain why. a. A square, given one diagonal b. A parallelogram, given two of its adjacent sides c. A rhombus, given its diagonals d. A parallelogram given a side and all the angles a. Calculate side =(Diagonal)/√2 Now side is known and by using side you can draw square b. Parallelogram cannot be drawn because you need an angle between the two adjacent sides also. c. Since diagonals are given and diagonals of a rhombus bisect each other perpendicularly. Now join the end points of the diagonals to get the rhombus. d. First draw the side of required length. Then draw the lines at the end points with the given angles. Now draw the angle bisector of each of these angles you will get two intersection points. Join these points to get the parallelogram. Assessment 12-4B: Exercises 2 & 11 2. Which of the following are always similar? Why? a. Any two rectangles in which the diagonal in one is twice as long as in the other. -Yes. The ratio of the side lengths is consistent, and so the rectangles are similar. – b. Any two rhombuses- Any two rhombuses= NO. same explanation as rectangle. rhombus 1 can have higher height and rhombus 2 can be wider. c. Any two circles because the corresponding parts like radius and diameter are proportional- YES. big circle or small circle. all are similar. d. Any two regular polygons- Polygons that are similar have all corresponding sides in the same proportion. This means they have the same shape, but can be different sizes e. Any two regular polygons with the same number of sides- They are always similar, regardless of size. Since they have the sides all the same length, they must always be in the same proportions, and so are always similar. The apothems and radii are in the same proportions as each other and the sides. 11. To find the height of a tree, a group of Girl Scouts devised the following method. A girl walks toward the tree along its shadow until the shadow of the top of her head coincides with the shadow of the top of the tree. If the girl is 150 cm tall, her distance to the foot of the tree is 15 m, and the length of her shadow is 3 m, how tall is the tree? The tree, its shadow and a line connecting the top of the tree to the tip of the shadow forms a right triangle that is similar to the right triangle formed by the girl, her shadow and a line connecting her head to the tip of her shadow. The smaller triangle has legs 1.5 m and 3 m. The larger has legs x m and (15+3) m, where x is the height of the tree. The ratios must be equal. So 1.5/3 = x/(15+3) ==> x = 18(1.5)/3 = 9m. Mathematical Connections 12-1 Exercise 9 9. a. Find at least five examples of congruent objects. b. Find at least five examples of similar objects that are not congruent. Mathematical Connections 12-2 Exercise 7 7. On a 4 by 4 geoboard or dot paper (with 16 dots), pose and solve questions concerning the congruence of the following: a. Triangles. b. Isosceles trapezoids. c. Quadrilaterals that are neither squares nor rectangles. Mathematical Connections 12-3 Exercises 2 & 3 2. Write a letter from you, a curriculum developer, to parents explaining whether or not the geometry curriculum in Grades 5–8 should include construction problems that use only a compass and straightedge. 3. c Chapter 14 Assessment 14-1B: Exercises: 4, 5, & 6 4. Find the coordinates of the points whose images under the translation (x, y) —› (x‒3, y+4) are the following: a. (7, 14) b. (¯ 7, ¯ 10) c. (b, k) 5. Consider the translation (x, y) —› (x+3, y-4). In each of the following: draw the image of the figure under the translation, and find the coordinates of the images of the labeled points: 6. Consider the translation(x, y) —› (x+3, y-4). In the following, draw the figure whose image is shown: Assessment 14-2B: Exercises 1 & 7 In the following figure, find the images of ΔABC under a reflection in l : 7. In the Beetle Bailey cartoon strip, Sarge’s pet’s name is Otto. a. If Otto is spelled in all capital letters, draw a reflecting line to show that the name can be its own image in a reflection. b. Where is the reflecting line? Assessment 14-3B: Exercises 1 & 5 In the following figures, describe a sequence of isometries followed by a size transformation so that the larger triangle is the final image of the smaller one: 5. Find the image of Δ ABC under the size transformation with center O and scale ½ . Assessment 14-4B Exercises 4 & 5 4. Determine the types of symmetry that each separate quilt pattern below has (line, rotational, point), if any. 5. Redesign the quilt pattern below so that it has line symmetry. Assessment 14-5B: Exercise 1 1. On dot paper, draw a tessellation of the plane using the following figure: Mathematical Connections 14-1 Exercise 2 a. If you rotate an object 180° clockwise or counter clockwise, using the same center, is the image the same in both cases? Explain. b. Answer part (a) if you rotate the object 360°. Mathematical Connections 14-3 Exercise 2 2. Given two similar figures, explain how to tell if there is a size transformation that transforms one of the figures onto the other.

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