Get Ready BLM... Solving Equations. Solve each equation. a) 4x + = 8y 5 = 6y + 7 c) z+ = z+ 5 d) d = 5 5 4. Write each equation in the form y = mx + b. a) x y + = 0 5x + y 7 = 0 c) x + 6y 8 = 0 d) 5 0 x y+ = Slope of a Line. Find the slope of the line through each pair of points. a) ( 4, 6) and ( 6, 0) (5, ) and (8, ) c) (, 8) and (5, ) d) (7, ) and (, 5) Equation for a Line 4. Find an equation for the line that a) has slope 4 and y-intercept 7 has slope and y-intercept 4 c) has slope 5 and passes through (4, ) d) has slope and passes through (, 5) 5. Find an equation for the line that passes through each pair of points. a) (, ) and (5, ) (, 4) and (, ) c) ( 5, ) and (, 4) d) (, 0) and (, ) Parallel and Perpendicular Lines 6. Find the slope of a line with each property. a) parallel to the line defined by y = 5x + 4 parallel to the line defined by y = 4 x c) perpendicular to the line defined by y = x + 5 d) perpendicular to the line defined by y = 5 x 7. Find an equation for the line that a) is parallel to the line defined by y = x + and passes through the point (4, 5) is parallel to the line defined by y = x + and passes through the point (, ) c) is perpendicular to the line defined by y = x + and passes through the point (, 5) d) is perpendicular to the line defined by y = x 5 and passes through the point 4 (, 4) Similar and Congruent Triangles 8. ABC is similar to PQR. a) Find the measure of P. Find the length of QR. BLM Get Ready
Section. Practice Master BLM... (page ). Determine the coordinates of the midpoint of each line segment. a). Determine the coordinates of the midpoint of the line segment defined by each pair of endpoints. a) (, 5) and (7, ) ( 4, ) and (5, ) c) (., 4.) and (5.6,.), 4, 5 4 5 4 d) ( ) and ( ). Find the slope of each median shown. a) c) 4. The endpoints of the diameter of a circle are A( 5, ) and B(, 7). Find the coordinates of the centre of this circle. BLM Section. Practice Master
5. The vertices of PQR are P(, 5), Q(5, 7), and R(, ). a) Find an equation in slope y-intercept form for the median from vertex P. Find an equation in slope y-intercept form for the median from vertex Q. c) Find an equation in slope y-intercept form for the median from vertex R. 6. One endpoint of a diameter of a circle centred at the origin is ( 5, ). Find the coordinates of the other endpoint of this diameter. 7. Determine an equation for the right bisector of the line segment with endpoints D(, 5) and M(7, 9). BLM... (page ) 9. Write an expression for the coordinates of the midpoint of the line segment with endpoints A(a, and B(4a, 5. Explain your reasoning. 0. a) Draw ABC with vertices A( 8, 0), B(0, 0), and C(0, 8). Construct the midpoints of AB, BC, and AC and label them D, E, and F, respectively. c) Join the midpoints to form DEF. d) Show that the length of line segment DE is one half the length of line segment AC. e) Show that the length of line segment DF is one half the length of line segment BC. f) Show that the length of line segment EF is one half the length of line segment AB. 8. a) Draw JKL with vertices J( 6, 4), K( 4, 5), and L(6, ). Draw the median from vertex J. Then, find an equation in slope y-intercept form for this median. c) Draw the right bisector of KL. Then, find an equation in slope y-intercept form for this right bisector. BLM Section. Practice Master
Section. Practice Master BLM 4.... Estimate the length of each line segment from its graph. Then, calculate its exact length. a) 5. The vertices of XYZ are X( 6, 8), Y(, 4), and Z(4, 6). a) Determine the exact length of each side of this triangle. Classify the triangle. c) Determine the perimeter of the triangle. Round your answer to the nearest tenth of a unit. 6. a) Show that the triangle with vertices D(, 0), E(0, 4), and F(, 0) is isosceles. List the coordinates of another isosceles triangle. 7. a) Determine the length of the median from vertex A in the triangle with vertices A( 6, 5), B(, 8), and C(4, 4). Describe how you could use geometry software to verify your answer to part a).. Calculate the length of the line segment defined by each pair of endpoints. a) ( 5, 6) and (, ) ( 7, 5) and ( 4, 6) c) ( 4.7,.8) and (6.4, 5.4),, 4 5 4 5 d) ( ) and ( ). A circle has a diameter with endpoints R( 4, 6) and T(0, 8). a) Find the length of this diameter. Find the length of the radius of this circle. 8. a) Determine the area of the right triangle with vertices D(, ), E(, ), and F(, 6). Describe how you could use geometry software to verify your answer to part a). 9. A line segment has endpoints S( 4, 5) and T(0, 7). a) Find the coordinates of the midpoint of line segment ST. Verify your answer to part a) by determining the distance from the midpoint to each of the endpoints and the distance between the endpoints. 4. For the triangle with vertices A(, 5), B(5, ), and C(, 5), determine the length of a) the median from A the median from B c) the median from C BLM 4 Section. Practice Master
Section. Practice Master BLM 6.... Find an equation for the line containing line segment CD. 6. In ABC, D is the midpoint of AB and E is the midpoint of AC.. A triangle has vertices D(, ), E(4, ), and F(6, 4). a) Draw DEF. Use analytic geometry to verify that DEF is a right angle.. Find the length of the median from vertex S. a) Find the coordinates of D and E. Show that DE is parallel to BC. c) Show that DE is half the length of BC. 7. The coordinates of the vertices of a triangle are D( 5, ), E(, 5), and F(, ). a) Draw DEF. Classify DEF. 8. Determine the shortest distance from a) the point (6, ) to the line y = x + the point ( 5, ) to the line y = x + c) the point (4, 5) to the line joining C(, ) and D(6, 4) 4. A quadrilateral has vertices P(, ), Q(5, 4), R(4, ), and S(, ). a) What type of quadrilateral is PQRS? Explain. Determine the perimeter of PQRS. Round your answer to the nearest tenth of a unit. 5. The endpoint of a radius of a circle with centre C(, ) is D(5, 5). Determine a) the length of the radius of the circle the coordinates of the endpoint E of the diameter DE of the circle 9. The points W(, ), X( 6, ), and Y(, 5) are three vertices of parallelogram WXYZ. a) Find the coordinates of vertex Z. Find the length of the diagonals XZ and WY. c) Show that the diagonals XZ and WY bisect each other. 0. A triangle has vertices A( 4, ), B(, 6), and C(6, ). a) Determine the length of the median from vertex A. Determine an equation in the form y = mx + b for the median from vertex A. BLM 6 Section. Practice Master
Section.4 Practice Master BLM 8.... Determine an equation for each circle. a) 5. Determine an equation for the circle that has a diameter with endpoints B( 4, 7) and C(4, 7). 6. The point A(4, lies on the circle defined by x + y = 5. a) Find the possible value(s) of b. Use a graph to show that the point(s) corresponding to the possible value(s) of b are on the circle. 7. a) Graph the circle defined by x + y =. Verify algebraically that the points M(, ) and N(, ) are on the circle. c) Find an equation in the form y = mx + b for the right bisector of chord MN. 8. a) Graph the circle defined by x + y = 45. Verify algebraically that the line segment joining P(, 6) and Q(6, ) is a chord of this circle. c) Find an equation in the form y = mx + b for the right bisector of chord PQ.. State the radius of the circle defined by each equation and give the coordinates of one point on the circle. a) x + y = 49 x + y = 6 c) x + y = 64 d) x + y =.44. Find an equation for the circle centred at the origin that passes through each point. a) (, 4) ( 5, ) c) (, 7) d) ( 6, ) 9. a) Graph the circle defined by x + y = 00. Verify algebraically that the point D(6, 8) lies on this circle. c) Construct the line segment DO. Determine the slope of the radius DO. d) Draw the line that is perpendicular to the line segment DO through the point D. Determine the slope of this line. e) Determine an equation for the tangent line in part d). 4. Determine whether each point is on, inside, or outside the circle defined by x + y = 6. a) (, ) ( 4, 6) c) (, 5) BLM 8 Section.4 Practice Master
Chapter Review BLM 9... (page ). Midpoint of a Line Segment. Find the midpoint of each line segment. a). Length of a Line Segment 4. Determine the length of the line segment defined by each pair of points. a) R( 5, 6) and S(, 6) T(4, 5) and U(4, 5) c) M( 5, 6) and N(, 4) d) P(, 6) and Q(7, ) 5. a) Determine the length of the median from vertex R of PQR. Determine the perimeter of PQR. Round your answer to the nearest tenth of a unit.. a) Determine the midpoint of the line segment with endpoints E( 6, 7) and F(, ). Determine the midpoint of the line segment with endpoints E( 5, 9) and F(, 4). 6. a) Draw the triangle with vertices X(, 4), Y(, ), and Z(, 6). Use analytic geometry to show that XYZ = 90. c) Determine the area of XYZ.. a) Draw the triangle with vertices A( 5, ), B(, 4), and C(, ). Draw the median from vertex A. Then, find an equation in the form y = mx + b for this median. c) Draw the right bisector of AC. Then, find an equation in the form y = mx + b for this right bisector. d) Draw the altitude from vertex C. Then, find an equation in the form y = mx + b for this altitude.. Apply Slope, Midpoint, and Length Formulas 7. Show that the triangle with vertices P(, 0), Q(0, ), and R(, 0) is equilateral. BLM 9 Chapter Review
8. a) Show algebraically that this triangle is isosceles. BLM 9... (page ) Find the midpoints of the equal sides. c) Show algebraically that the line segment joining the midpoints of the equal sides is parallel to the third side of the triangle. c) 9. On a map, a ski hill has a chair lift running straight from A(0, 5) to B(60, 55). a) How long is the section of the chair lift if each unit on the map grid represents m, to the nearest tenth of a metre? Is the point C(50, 45) on the chair lift? Explain your reasoning..4 Equation for a Circle 0. Determine an equation for each circle. a). Find an equation for the circle that is centred at the origin and a) has a radius of.7 has a radius of 8 c) has a diameter of 8 d) passes through the point (, 5). a) Show that the line segment joining C(, 5) and D( 5, ) is a chord of the circle defined by x + y = 9. Determine an equation for the right bisector of the chord CD.. a) Show that point B(, ) lies on the circle defined by x + y =. Find an equation for the radius from the origin O to point B. c) Find an equation for the line that passes through B and is perpendicular to OB. BLM 9 Chapter Review
Chapter Practice Test BLM.... The midpoint of the line segment with endpoints A( 4, 5) and B(, ) is A (, 4) B (, ) C (, ) D ( 4.5,.5) 8. The library is located exactly halfway between Brandon s house and Vaughn s house. The intervals on the grid represent km.. The length of the line segment with endpoints C(, 5) and D(, 4) is A 8 B 40 C 06 D 6. An equation for the circle with centre (0, 0) and radius 8 is A x + y = 64 B x + y = 6 C x + y = 8 D x + y = 4. The endpoints of a diameter of a circle are A(, 7) and B(5, ). The coordinates of the centre of this circle are A ( 4, 5) B (, ) C (, ) D (, 7) 5. The point ( 4, 5) lies on a circle with centre (0, 0). An equation for the circle is A x + y = 0 B x + y = 9 C x + y = D x + y = 4 6. Find the midpoint and the length of the line segment defined by each pair of endpoints. a) A( 9, ) and B(5, 4) C(, 5) and D(5, ) 7. a) Draw the triangle with vertices A( 5, ), B(, 6), and C(, ). Determine an equation for the median from A. c) Determine an equation for the perpendicular bisector of AB. a) How far apart are Brandon s house and Vaughn s house, to the nearest tenth of a kilometre? Determine the coordinates of the library. 9. The vertices of a triangle are D( 4, ), E(, 6), and F(6, 4). a) Determine the lengths of the sides of the triangle. Classify DEF. Explain your reasoning. c) Determine the perimeter of DEF. Round your answer to the nearest tenth of a unit. d) Describe how you could use geometry software to verify your answers in parts a),, and c). 0. a) Plot the triangle with vertices G( 5, 4), H(, 8), and I(, 6). Determine an equation for the median from vertex G. c) Determine an equation for the right bisector of GH. d) Determine an equation for the altitude from G to HI. BLM Chapter Practice Test
Chapter Test BLM.... The midpoint of the line segment with endpoints A(, 5) and B(7, ) is A (, ) B ( 5, ) C (4, 4) D (, ) 8. A computer store is located exactly halfway between David s house and his school. The intervals on the gridlines represent km.. The length of the line segment with endpoints C( 5, ) and D(, ) is A 7 B C 65 D 7. An equation for the circle with centre (0, 0) and radius 6 is A x + y = 6 B x + y = 9 C x + y = 6 D x + y = 4. The endpoints of a diameter of a circle are A( 4, ) and B(, 5). The coordinates of the centre of this circle are A (, 4) B (, ) C (, 4) D (, ) 5. The point (6, ) lies on a circle with centre (0, 0). The equation of the circle is A x + y = 45 B x + y = C x + y = 9 D x + y = 8 6. Find the midpoint and the length of the line segment defined by each pair of endpoints. a) A( 5, 4) and B(, 6) C( 4, ) and D(, ) 7. a) Draw the triangle with vertices A( 5, ), B(, 5), and C(, ). Determine an equation in slope y-intercept form for the median from B. c) Determine an equation for the perpendicular bisector of AB. a) How far apart are David s house and his school, to the nearest tenth of a kilometre? Determine the coordinates of the computer store. 9. The vertices of a triangle are D( 4, 5), E( 7, ), and F(, ). a) Determine the lengths of the sides of the triangle. Classify DEF. Explain your reasoning. c) Determine the perimeter of DEF. Round your answer to the nearest tenth of a unit. d) Describe how you could use geometry software to verify your answers to parts a),, and c). 0. a) Plot the triangle with vertices X( 4, ), Y(, 5), and Z(4, ). Determine an equation in slope y-intercept form for the median from vertex X. c) Determine an equation for the right bisector of XY. d) Determine an equation for the altitude from Y to XZ. BLM Chapter Test
BLM Answers BLM 4... (page ) Get Ready. a) x = y = 6 c) z = 40 d) d = 5. a) y = x + y = 5x + 7 4 c) y = x+ d) y = x+ 5 5. a) c) 8 5 d) 4. a) y = 5x 7 7 y = x+ c) y = 4x + 7 d) y = x 4 5. a) y = x + y = x + c) y = x d) y = x 6. a) m = 5 m = 4 c) m = d) m = 5 7. a) y = x y = x+ 6 c) 6 4 4 y = x d) y = x 8. a) 60 5 cm Section. Practice Master. a) (, ). a) (4, 4) (, ) c) (., 0.9) d) (,0 5 ) (, ) c) (, ) 6 9. a) m = m = 5 4. (, ) 6 6 5. a) y = x+ y = x+ 7 7 5 9 c) y = x+ 6. (5, ) 5 4 7. y = x 7 7 8. a),, c) 9. (a, 4; Use the midpoint formula with x = a, x = 4a, y = b, and y = 5b. 0. a),, c) d) From the graph, DE is 4 units long and AC is 8 units long, so DE is one half the length of AC. e) From the graph, DF is 4 units long and BC is 8 units long, so DF is one half the length of BC. f) By the Pythagorean theorem, EF = DF + DE AB = AC + BC = 4 + 4 = 8 + 8 = = 8 EF = AB = 8 = 6 = 64 = 6 = 64 = 4 = 8 Line segment AB is twice the length of line segment EF. Chapter Practice Masters Answers
Section. Practice Master. Estimates may vary. Exact answers: a) 9 58. a) 8 0 c) 07.85 d). a) 9 98 4. a) 7 45 c) 9 5. a) XY = 60, XZ = 04, YZ = 6 scalene c) 4.5 6. a) DE = 5, EF = 5, DF = 6; DE = EF, so DEF is isosceles. Answers may vary. For example: A( 5, 0), B(0, 0), C(5, 0) 7. a) 58 Construct ABC by plotting the vertices and connecting them with line segments. Construct the midpoint, D, of side BC. Construct line segment AD. Select line segment AD and measure its length. 8. a) 5 square units Construct DEF by plotting the vertices and connecting them with line segments. Select the vertices and then construct the triangle interior. Select the triangle interior and measure its area. 9. a) M(, ) SM = 85, MT = 85 ST = 40 = 4 85 = 4 85 = 85 = SM = MT So, ST = SM = MT. M is the midpoint of ST. Section. Practice Master. y = x. a) BLM 4... (page ) 5. a) E(, ) 6. a) D(, 5), E(, ) slope DE = ; slope BC = ; since the slopes are equal, the line segments are parallel. c) DE = 0 BC = 80 = 4 0 = 4 0 = 0 Thus, DE is half the length of BC. 7. a) DEF is isosceles. 8. a) 9. c) 6.5 9. a) Z(6, ) XZ = 45, WY = 65 c) The midpoint of XZ and the midpoint of WY both occur at (0,.5). Thus, XZ and WY bisect each other. 0. a) 7 y = x Section.4 Practice Master. a) x + y = 6 x + y = 7. Points may vary. Examples are given. a) 7; (0, 7) 4; (4, 0) c) 8; ( 8, 0) d).; (0,.). a) x + y = 5 x + y = 9 c) x + y = 58 d) x + y = 40 4. a) inside outside c) on 5. x + y = 65 6. a), slope DE = ; slope EF = ; since the slopes are negative reciprocals, DE is perpendicular to EF, and DEF is a right angle.. 65 4. a) PQRS is a rhombus, because all four sides are equal in length. 4. units Chapter Practice Masters Answers
7. a) M(, ): L.S. = x + y R.S. = = ( ) + = 9+ 4 = N(, ): L.S. = x + y R.S. = = + ( ) = 4+ 9 = c) y = x 8. a) BLM 4... (page ) Check that both endpoints are on the circle. P(, 6): L.S. = x + y R.S. = 45 = ( ) + 6 = 9+ 6 = 45 Q(6, ): L.S. = x + y R.S. = 45 = 6 + ( ) = 6 + 9 = 45 c) y = x 9. a), c), d) D(6, 8) L.S. = x + y R.S. = 00 = 6 + ( 8) = 6 + 64 = 00 4 c) d) 5 e) y = x 4 4 Chapter Review. a) (, ) (, ). a) ( 4, 4) (, 5) Chapter Practice Masters Answers
. a),, c), d) 5 y = x c) d) y = x+ 4. a) 0 c) 64 d) 6 5. a) 5.8 6. a) y = 8x BLM 4... (page 4). a) x + y =.69 x + y = 8 c) x + y = 8 d) x + y = 4. a) Check that both endpoints are on the circle. C(, 5): L.S. = x + y R.S. = 9 = ( ) + 5 = 4+ 5 = 9 D( 5, ): L.S. = x + y R.S. = 9 = ( 5) + = 5 + 4 = 9 y = x. a) Check that the point B(, ) satisfies the equation x + y =. L.S. = x + y R.S. = = ( ) + ( ) = 9+ 4 = y = x c) y = x slope XY = ; slope YZ = ; since the slopes are negative reciprocals, XYZ = 90. c) 6 square units 7. PQ = QR = PR = ; all three sides have equal length, so PQR is equilateral. 8. a) DE = EF = 6 ; since two sides have equal length, DEF is isosceles. G (, ) is the midpoint of DE; H 7 (, ) is the midpoint of EF. c) slope GH = ; slope DF = ; since the slopes are equal, the two segments are parallel. 9. a) 4.4 m Yes. The line y = x 5 contains the points A and B. Since it also contains the point C, and C is between A and B, C is on the chair lift. 0. a) x + y = 49 x + y = 7 c) x + y = 6.5 Chapter Practice Test. C. D. A 4. B 5. D 6. a) (, ); 00 (, 7) 7. a) ; 58 7 y = x+ c) y = x+ 4 4 8. a) 7. km (5, 5) Chapter Practice Masters Answers
9. a) DE = 68 ; EF = 64 ; DF = 04 DEF is scalene because the three sides have different lengths. c). units d) Construct DEF by plotting the vertices and connecting them with line segments. Select the sides of the triangle and then measure their lengths. Select the vertices of the triangle and then construct the triangle interior. Select the triangle interior and measure the perimeter. 0. a) 7. a) BLM 4... (page 5) 7 y = x+ c) y = x 4 8. a) 0. km (6, 5) 9. a) DE = 45 ; EF = 6; DF = 45 isosceles; DE = DF c) 9.4 units d) Construct DEF by plotting the vertices and connecting them with line segments. Select the sides of the triangle and then measure their lengths. Select the vertices of the triangle and then construct the triangle interior. Select the triangle interior and measure the perimeter. 0. a) 5 y = x+ c) 6 6 y = x+ d) 8 y = x 7 7 Chapter Test. A. D. C 4. B 5. A 6. a) (, ); 64 (, 5 ) ; 6 6 9 y = x+ c) 5 5 y = x+ d) y = x + 4 4 Chapter Practice Masters Answers