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Name _________________________________________ Geometry – Pd __ Date ____________ Constructions Constructions 1. Congruent Line Segments Given: (Line segment) AB Task: To construct a line segment congruent to (line segment) AB 1. If a reference line does not already exist, draw a reference line with your straightedge. Place a starting point on the reference line. 2. Place the point of the compass on point A. 3. Stretch the compass so that the pencil is exactly on B. 4. Without changing the span of the compass, place the compass point on the starting point on the reference line and swing the pencil so that it crosses the reference line. Label your copy. Your copy and (line segment) AB are congruent. Congruent means equal in length. Example 1) Example 2) 2. Congruent Angles Given: Regents Common Core BAC Task: To construct an angle congruent to BAC 1. If a reference line does not already exist, draw a reference line with your straightedge. Place a starting point on the reference line. 2. Place the point of the compass on the vertex of BAC (point A). 3. Stretch the compass to any length so long as it stays ON the angle. 4. Swing an arc with the pencil that crosses both sides of BAC . 5. Without changing the span of the compass, place the compass point on the starting point of the reference line and swing an arc that will intersect the reference line and go above the reference line. 6. Go back to BAC and measure the width (span) of the arc from where it crosses one side of the angle to where it crosses the other side of the angle. 7. With this width, place the compass point on the reference line where your new arc crosses the reference line and mark off this width on your new arc. 8. Connect this new intersection point to the starting point on the reference line. Example 1) Example 2) 3. Construct an Angle Bisector Given: Regents Common Core BAC Task: Bisect BAC 1. Place the point of the compass on the vertex of BAC (point A). 2. Stretch the compass to any length so long as it stays ON the angle. 3. Swing an arc so the pencil crosses both sides of BAC . This will create two intersection points with the sides (rays) of the angle. 4. Place the compass point on one of these new intersection points on the sides of BAC . If needed, stretch your compass to a sufficient length to place your pencil more than halfway into the interior of the angle. Stay between the sides (rays) of the angle. Place an arc in this interior - you do not need to cross the sides of the angle. 5. Without changing the width of the compass, place the point of the compass on the other intersection point on the side of the angle and make the same arc. Your two small arcs in the interior of the angle should be crossing. 6. Connect the point where the two small arcs cross to the vertex A of the angle. Example 1) Example 2) Name _________________________________________ Geometry – Pd __ Date ____________ Constructions Constructions HW Μ Μ Μ Μ , construct a congruent segment. 1) Using π΄π΅ [Leave all construction marks] 2) Construct an angle congruent to < π΄π΅πΆ [Leave all construction marks] 3) Create an angle bisector for < ππ½π». [Leave all construction marks] 4) Using a compass, construct the bisector of < ABC. [Leave all construction marks] _____5) When given a square, the construction of an angle bisector at any vertex will create the diagonal of the square. (1) True (2) False _____ 6) Which illustration shows the correct construction of an angle bisector? (1) (2) (3) (4) _____ 7) Based on the construction to the right, which statement must be true? 1 (1) π < π΄π΅π· = π < πΆπ΅π· 2 (2) π < π΄π΅π· = π < πΆπ΅π· (3) π < π΄π΅π· = π < π΄π΅πΆ 1 (4) π < πΆπ΅π· = 2 π < π΄π΅π· _____ 8) As shown in the diagram, a compass is used to find points D and E, equidistant from point A. Next the compass is used to find point F, equidistant from point D and E. Finally, a straight edge is used to draw AF. Which statement must be true? Μ Μ Μ Μ bisects side π΅πΆ Μ Μ Μ Μ (1) π΄πΉ Μ Μ Μ Μ (2) π΄πΉ bisects < π΅π΄πΆ Μ Μ Μ Μ ⊥ π΅πΆ Μ Μ Μ Μ (3) π΄πΉ (4) β ABG ~β ACG _____ 9) The diagram shows the construction of the bisector of < π΄π΅πΆ. Which statement is not true? 1 (1) π < πΈπ΅πΉ = π < π΄π΅πΆ 2 1 (2) π < π·π΅πΉ = π < π΄π΅πΆ 2 (3) π < πΈπ΅πΉ = π < π΄π΅πΆ (4) π < π·π΅πΉ = π < πΈπ΅πΉ _____ 10) A straight edge and compass were used to create the following construction. Which statement is false? (1) π < π΄π΅π· = π < π·π΅πΆ 1 (2) (π < π΄π΅πΆ) = π < π΄π΅π· 2 (3) 2(π < π·π΅πΆ) = π < π΄π΅πΆ (4) 2(π < π΄π΅πΆ) = π < πΆπ΅π· Review Section!! _____ 11) In the diagram, transversal TU intersects PQ and RS at V and W respectively. If m<TVQ = 5x – 22 and m<VWS = 3x + 10, for which value of x is PQ//RS? (1) 6 (2) 16 (3) 24 (4) 28 _____ 12) When the system of equations π¦ + 2π₯ = π₯ 2 and π¦ = π₯ is graphed on a set of axes, what is the total number of points of intersection? (1) 0 (2) 1 (3) 2 (4) 3 _____ 13) The equation of a line is 3π¦ + 2π₯ = 12. What is the slope of the line perpendicular to the given line? 3 2 (1) (2) 2 (3) 3 −2 3 2 (4) − 3 Constructions HW Answers 1) 2) 3) 4) 5) (1) 6) (3) 7) (2) 8) (2) 9) (3) 10) (4) 11) (2) 12) (3) 13) (1) Review Section!! 11) In the diagram, transversal TU intersects PQ and RS at V and W respectively. If m<TVQ = 5x – 22 and m<VWS = 3x + 10, for which value of x is PQ//RS? 12) When the system of equations π¦ + 2π₯ = π₯ 2 and π¦ = π₯ is graphed on a set of axes, what is the total number of points of intersection? 13) The equation of a line is 3π¦ + 2π₯ = 12. What is the slope of the line perpendicular to the given line? Name _________________________________________ Geometry – Pd __ Date ____________ Constructions Constructions Day 2 1. Construct an Equilateral Triangle Regents Common Core Given: (Line segment) AB Task: To construct an equilateral triangle 1. Place the point of the compass on point A and measure (place the pencil) to point B. Swing a large arc up above the given line. 2. Place the point of the compass on point B and measure (place the pencil) to point A. Swing a large arc up above the given line. 3. Create the missing two sides of the equilateral triangle by connecting point A to the point of intersection, and then point B to the point of intersection. Example 1) Example 2) Common Core Application: 1) Margie has three cats. She has heard that cats in a room position themselves at equal distances from one another and wants to test that theory. Margie notices that Simon, her tabby cat, is in the center of her bed (at S), while JoJo, her Siamese, is laying on her desk chair (at J). If the theory is true, where will she find Mack, her calico cat? Use the scale drawing of Margie’s room shown below, together with (only) a compass and straightedge. Place an M where Mack will be if the theory is true. 2. Construct a ππ° Angle Regents Common Core Given: (Line segment) AB Task: To construct a 30° Angle 1. First, construct an equilateral triangle based on segment Μ Μ Μ Μ π΄π΅. a) Place the point of the compass on point A and measure to point B. Swing a large arc up above the given line. b) Place the point of the compass on point B and measure to point A. Swing a large arc up above the given line. c) Create the missing two sides of the equilateral triangle by connecting both point A and point B to the point of intersection 2. Second, bisect any angle of the equilateral triangle. a) Place the point of the compass on the vertex b) Stretch the compass to any length so long as it stays ON the angle c) Swing an arc so the pencil crosses both sides of the angle d) Place the compass point on one of these new intersection points on the sides of the angle and swing an arc (using the pencil) on the interior of the angle. e) Without changing the width of the compass, complete the same for the other point of intersection. f) Connect the point where the two small arcs cross to the vertex of the angle. 3. You have now created the construction of a 30° angle. Name _________________________________________ Geometry – Pd __ Date ____________ Constructions Constructions Day 2 HW Μ Μ Μ Μ , construct an equilateral triangle. 1) Using π΄π΅ [Leave all construction marks] 2) Construct an angle bisector to < ππ½π» [Leave all construction marks] 3) β π΄π΅πΆ is shown below. Is it an equilateral triangle? Justify your response. 4) Using equilateral β π΄π΅πΆ to construct a 30° angle. [Leave all construction marks] _____ 5) One method of constructing an equilateral triangle is to simply construct a triangle using the same segment for each side. (1) True (2) False _____ 6) Which diagram shows the construction of a 30° angle? _____ 7) The diagram shows the construction of an equilateral triangle. Which statement justifies this construction? (1) < π΄ + < π΅ + < πΆ = 180 (2) < π΄ = < π΅ = < πΆ (3) π΄π΅ = π΅πΆ = π΄πΆ (4) π΄π΅ + π΅πΆ > π΄πΆ _____ 8) The diagram below shows the construction of the bisector of β‘ π΄π΅πΆ. Which statement is not true? 1 1) πβ‘ πΈπ΅πΉ = 2 πβ‘ π΄π΅πΆ 1 2) πβ‘ π·π΅πΉ = 2 πβ‘ π΄π΅πΆ 3) πβ‘ πΈπ΅πΉ = πβ‘ π΄π΅πΆ 4) πβ‘ π·π΅πΉ = πβ‘ πΈπ΅πΉ Review Section!! _____ 9) What is the equation of a line passing through the point (4, −1) and parallel to the line whose equation is 2π¦ − π₯ = 8? 1 1 (1) π¦ = 2 π₯ − 3 (2) π¦ = 2 π₯ − 1 (3) π¦ = −2π₯ + 7 (4) π¦ = −2π₯ + 2 _____ 10) In the diagram of βπππ shown below, PR is extended to S, m<P = 110, m<Q = 4x, and m<QRS = π₯ 2 + 5π₯. What is the m<Q? (1) 44 (2) 40 (3) 11 (4) 10 _____ 11) The solution of the system of equations π¦ = π₯ 2 − 2 and π¦ = π₯ is: (1) (1,1) and (−2, −2) (2) (2,2) and (−1, −1) (3) (1,1) and (2,2) (4) (−2, −2) and (−1, −1) _____ 12) Lines p and q are intersected by line r, as shown. If m<1 = 7x – 36 and m<2 = 5x + 12, for which value of x would p//q? (1) 17 (2) 24 (3) 83 (4) 97 Constructions Day 2 HW 1) 2) 3) 4) 5) (1) 6) (1) 7) (3) 8) (3) 9) (1) 10) (2) 11) (2) 12) (1) 9) What is the equation of a line passing through the point (4, −1) and parallel to the line whose equation is 2π¦ − π₯ = 8? 10) In the diagram of βπππ shown below, PR is extended to S, m<P = 110, m<Q = 4x, and m<QRS = π₯ 2 + 5π₯. What is the m<Q? 11) The solution of the system of equations π¦ = π₯ 2 − 2 and π¦ = π₯ is: 12) Lines p and q are intersected by line r, as shown. If m<1 = 7x – 36 and m<2 = 5x + 12, for which value of x would p//q? Name _________________________________________ Geometry – Pd __ Date ____________ Constructions Constructions Day 3 Recall: An ______________________ is a line segment with one endpoint on any vertex of a triangle that extends to the opposite side of the triangle and bisects the angle. Since there are three vertices in every triangle, there are ___________ angle bisectors of a triangle. The point of concurrency of the angle bisectors of a triangle is known as the ______________ of a triangle. To construct the incenter of a given triangle construct the angle bisector on ________ _______________________ vertices. The incenter will always be located ______________ a given triangle. The point of concurrency (the incenter) is the center of the circle that is inscribed within a given triangle. The Incenter Regents 1) Using a compass and a straightedge, construct the incenter of βπ΄π΅πΆ. Common Core The Incenter: - The incenter is formed by connecting the three angle bisectors - The three angle bisectors of a triangle are concurrent at a point equidistant from the sides of a triangle. Directions: Using the above information, complete the following questions. Don’t forget justifications. 1) The incenter of βπππ is located at point P. If CP = 4x + 9 and PB = 6x – 11, find the value of x and the length of CP and PD. Justify all calculations. 2) Point P is the incenter of βπΉπΊπ». If m<TFP = 3x + 15, and m<UFP = 5x – 13, find the value of x. Justify all calculations. 3) The incenter of βπΆπ·πΈ is point P. If m<SDP = 7x + 5 and m<UDP = 9x – 5, find the value of x and m<SDP. Justify all calculations. 4) P is the incenter of βπππ. If m<SZP = 7x + 7, and m<SZT = 16x + 4, find the value of x and m<SZT. Justify all calculations. Name _________________________________________ Geometry – Pd __ Date ____________ Constructions Constructions Day 3 HW 1) Construct the incenter of βπππ. 2) Construct an equilateral triangle to DE. _____ 3) P is the incenter of βπππ. If m<RYP = 2x + 20, and m<TYP = x + 40, what is the m<RYT? (1) 20 (2) 40 (3) 60 (4) 120 _____ 4) Which geometric principle is used in the construction shown below? (1) The intersection of the angle bisectors of a triangle is the center of the inscribed circle. (2) The intersection of the angle bisectors of a triangle is the center of the circumscribed circle. (3) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the inscribed circle. (4) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the circumscribed circle. _____ 5) The incenter of βπ΄π΅πΆ is located at point G. If EG = 3x + 14 and DG = 5x – 8, what is the length of GF? (1) 5 (2) 11 (3) 22 (4) 47 _____ 6) A straight edge and compass were used to create the following construction. Which statement is false? (1) π < π΄π΅π· = π < π·π΅πΆ 1 (2) 2 (π < π΄π΅πΆ) = π < π΄π΅π· (3) 2(π < π·π΅πΆ) = π < π΄π΅πΆ (4) 2(π < π΄π΅πΆ) = π < πΆπ΅π· _____ 7) The diagram shows the construction of an equilateral triangle. Which statement justifies this construction? (1) < π΄ + < π΅ + < πΆ = 180 (2) < π΄ = < π΅ = < πΆ (3) π΄π΅ = π΅πΆ = π΄πΆ (4) π΄π΅ + π΅πΆ > π΄πΆ Review Section!! _____ 8) What is the slope of a line perpendicular to the line whose equation is 3π₯ − 7π¦ + 14 = 0? 3 7 (1) 7 (2) − 3 1 (4) − 3 (3) 3 _____ 9) Line m and point P are shown in the graph. Which equation represents the line passing through P and parallel to line m? (1) π¦ − 3 = 2(π₯ + 2) (2) π¦ + 2 = 2(π₯ − 3) 1 (3) π¦ − 3 = − 2 (π₯ + 2) 1 (4) π¦ + 2 = − 2 (π₯ − 3) _____ 10) In βπ΄π΅πΆ, m<A = 3x + 1, m<B = 4x – 17, and m<C = 5x – 20. Which type of triangle is βπ΄π΅πΆ? (1) right (2) scalene (3) isosceles (4) equilateral _____ 11) Transversal EF intersects AB and CD as shown. Which statement could always be used to prove AB // CD? (1) < 2 ≅< 4 (2) < 3 and < 6 are supplementary (3) < 7 ≅< 8 (4) < 1 and < 5 are supplementary Constructions Day 3 HW 1) 2) 3) (4) 4) (1) 5) (4) 6) (4) 8) (2) 9) (2) 10) (3) 11) (2) 8) What is the slope of a line perpendicular to the line whose equation is 3π₯ − 7π¦ + 14 = 0? 9) Line m and point P are shown in the graph. Which equation represents the line passing through P and parallel to line m? 10) In βπ΄π΅πΆ, m<A = 3x + 1, m<B = 4x – 17, and m<C = 5x – 20. Which type of triangle is βπ΄π΅πΆ? 11) Transversal EF intersects AB and CD as shown. Which statement could always be used to prove AB // CD? (1) < 2 ≅< 4 (2) < 3 and < 6 are supplementary (3) < 7 ≅< 8 (4) < 1 and < 5 are supplementary 7) (3)